College Algebra Notes - Metropolitan Community College · PDF filex+ 3 + 2 x+ 2 x+ 3 Example 16. Simplify the complex rational expression. State any domain restrictions. 1 + 6 x +

College Algebra Notes

Joseph Lee

Metropolitan Community College

Contents

Introduction 2

Unit 1 3Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Polynomial, Radical, Rational, and Absolute Value Equations . . . . . . . . . . . . . . . . . . . . 12Linear and Absolute Value Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Extremum, Symmetry, Piecewise Functions, and the Difference Quotient . . . . . . . . . . . . . . 27Graphing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Unit 2 41Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47The Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Polynomial and Rational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Unit 3 65Operations on Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Properties of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Exponential and Logarithmic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Unit 4 88Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

1

Joseph Lee

Introduction

The question I receive most often, regardless of the course, is, “When am I ever going to use this?” I thinkthe question misses the point entirely. While I do not determine which classes students need to get theirdegree, I do think it is a good policy that students are required to take my course – for more reasons thanjust my continued employment, which I support as well.

If a student asked an English instructor why he or she had to read Willa Cather’s My Antonia, theinstructor would not argue that understanding nineteenth century prairie life was essential to becoming acompetent tax specialist or licensed nurse. The instructor would not argue that reading My Antonia wouldbenefit the student directly through a future application. Instead, the benefit of reading this beautiful pieceof American literature is entirely intrinsic. The mere enjoyment and appreciation is enough to justify itsplace in a post-secondary education. Moreover, the results arrived to througout the course are as beau-tiful as any prose or poetry a student will encounter in his or her studies here at Metro or any other college.

2

College Algebra Notes Joseph Lee

Unit 1

Rational Expressions

Domain of a Rational Expression

A rational expression will be defined as long as the denominator does not equal zero.

Example 1. State the domain of the rational expression.

x

x + 3


2x + 3

3x− 2


x− 4

x2 + 5x + 6


x2 + 8x + 7

x2 + 1

3


Example 5. Simplify. State any domain restrictions.

4x− 8

x− 2


x− 3

x2 − 5x + 6


x2 − 14x + 49

x2 − 6x− 7

4


Example 8. Multiply. State any domain restrictions.

2x

3x + 1· 3x2 − 5x− 2

4x2 + 8x

Example 9. Multiply. State any domain restrictions.

x2 + x− 6

4− x2· x

2 + 4x + 4

x2 + 4x + 3

Example 10. Divide. State any domain restrictions.

2x2 − 9x− 5

2x2 − 13x + 15÷ 4x2 − 1

4x2 − 8x + 3

5


Example 11. Add. State any domain restrictions.

x2 − 5x

x2 − 7x + 12+

3x− 3

x2 − 7x + 12

Example 12. Add. State any domain restrictions.

x− 1

x2 + x− 20+

x + 3

x2 − 5x + 4

6


Example 13. Subtract. State any domain restrictions.

4

x + 2− x− 26

x2 − 3x− 10

Example 14. Simplify the complex rational expression. State any domain restrictions.

1− 2

x

1− 4

x2

7



x

x + 3+ 2

x +2

x + 3


1 +6

x+

9

x2

1− 1

x− 12

x2

8


Quadratic Equations

Definition: Quadratic Equation

A quadratic equation is an equation that can be written as

ax2 + bx + c = 0

where a, b, and c are real numbers and a 6= 0.

Zero Factor Property

If a · b = 0, then a = 0 or b = 0.

Example 1. Solve.x2 − 5x + 6 = 0

Example 2. Solve.3x(x− 2) = 4(x + 1) + 4

Square Root Property

If x2 = a, then x = ±√a.

9


Example 3. Solve.3x2 + 4 = 58

Example 4. Solve.(x− 3)2 = 4

Example 5. Solve.(2x− 1)2 = −5

10


Quadratic Formula

For any quadratic equation ax2 + bx + c = 0,

x =−b±

√b2 − 4ac

2a

Example 6. Solve.3x2 − 5x− 2 = 0

Example 7. Solve.x2 − 3x− 7 = 0

11


Polynomial, Radical, Rational, and Absolute Value Equations

Example 1. Solve.x3 − 16x = 0

Example 2. Solve.8x3 + 6x = 12x2 + 9

Example 3. Solve.x + 1 =

√x + 13

12


Example 4. Solve. √x2 − x + 3− 1 = 2x

Example 5. Solve. √x− 1 =

√2x + 2

13


Example 6. Solve.(x− 1)2/3 = 4

Example 7. Solve.x6 − 6x3 + 9 = 0

14


Example 8. Solve.x−2 + 2x−1 − 15 = 0

Example 9. Solve.8

(x− 4)2− 6

x− 4+ 1 = 0

15


Definition. Absolute Value.

The absolute value of a real number x is the distance between 0 and x on the real number line.The absolute value of x is denoted by |x|.

Example 10. Solve.|x| = 7

Observation 1

For any nonnegative value k, if |x| = k, then x = k or x = −k.

Example 11. Solve.|x− 3| = 2

Example 12. Solve.|3x + 5| = 8

16


Example 13. Solve.|x + 2| = −3

Observation 2

For any negative value k, the equation |x| = k has no solution.

Example 14. Solve.|x + 9| − 3 = 1

Example 15. Solve.−2|3x + 2|+ 1 = 0

17


Linear and Absolute Value Inequalities

Definition: Union and Intersection

Let A and B be sets.

The union of A and B, denoted A ∪ B is the set of all elements that are members of A, or B, orboth.

The intersection of A and B, denoted A ∩ B is the set of all elements that are members of bothA and B.

Example 1. Let A = {1, 2, 3} and B = {2, 4, 6}. Determine both A ∪B and A ∩B.

A ∪B =

A ∩B =

Example 2. Let B = {2, 4, 6} and C = {1, 3, 5}. Determine both B ∪ C and B ∩ C.

B ∪ C =

B ∩ C =

Example 3. Let D = {x | 0 < x < 4} and E = {x | 2 < x < 6}. Determine both D ∪ E and D ∩ E.

D ∪ E =

D ∩ E =

Interval Notation

For any real numbers a and b, the following are sets written in interval notation.

(a, b) = {x | a < x < b}

(a, b] = {x | a < x ≤ b}

[a, b) = {x | a ≤ x < b}

[a, b] = {x | a ≤ x ≤ b}

18


Example 4. Write the following sets in interval notation.

{x | − 3 ≤ x < 5} =

{x | 7 < x ≤ 10} =

Example 5. Write the following sets in set-builder notation.

(3, 8) =

[−2, 5] =

Unbounded Intervals(a,∞) = {x |x > a}

[ a,∞) = {x |x ≥ a}

(−∞, b) = {x |x < b}

(−∞, b ] = {x |x ≤ b}

Example 6. Write the following sets in interval notation.

{x |x ≥ −2} =

{x |x < −2} =

Example 7. Let A = (1, 4) and B = (2, 5). Determine both A ∪B and A ∩B.

A ∪B =

A ∩B =

Example 8. Let B = (2, 5) and C = [3, 6]. Determine both B ∪ C and B ∩ C.

B ∪ C =

B ∩ C =

19


Example 9. Let D = (0, 4] and E = [5, 9). Determine both D ∪ E and D ∩ E.

D ∪ E =

D ∩ E =

Example 10. Solve.3x− 7 < 5

Solution:

Example 11. Solve.−2x− 7 ≤ 19

Solution:

Example 12. Solve.1 < 4x− 3 ≤ 11

Solution:

20


Example 13. Solve.

−2 ≤ 1− 2x

3≤ 3

Solution:

Example 14. Solve.|x| < 4

Solution:

Observation 3

For any nonnegative value k, the inequality |x| < k may be expressed as

−k < x < k.

Similarly, for |x| ≤ k, we have −k ≤ x ≤ k.

Example 15. Solve.|x| > 4

Solution:

Observation 4

For any nonnegative value k, the inequality |x| > k may be satisfied by either

x > k or x < −k.

Similarly, for |x| ≥ k, we know x ≥ k or x ≤ −k.

21


Example 16. Solve.|x + 8| ≤ 2

Solution:

Example 17. Solve.|6x + 2| ≥ 2

Solution:

Example 18. Solve.|4− x| < 8

Solution:

22


Example 19. Solve.|1− 7x| > 13

Solution:

Example 20. Solve.|x− 3| > −2

Solution:

Observation 5

For any negative value k, the inequality |x| > k holds for any value of x.

Example 21. Solve.|3x + 2| < −5

Solution:

Observation 6

For any negative value k, the inequality |x| < k has no solution.

23


Functions

Definition: Relation

A relation is a correspondence between two sets. Elements of the first set are called the domain.Elements of the second set are called the range.

Definition: Function

A function is a specific type of a relation where each element in the domain corresponds to ex-actly one element in the range.

Example 1. Determine the domain and range of the following relation1. Does the relation define afunction?

{(Joseph, turkey), (Joseph, roast beef), (Michael,ham)}

Domain:

Range:

Function?

Example 2. Determine the domain and range of the following relation. Does the relation define afunction?

{(1, 3), (2, 4), (−1, 1)}

Domain:

Range:

Function?

Example 3. Determine the domain and range of the following relation. Does the relation define afunction?

{(3, 5), (4, 5), (5, 5)}

Domain:

Range:

Function?

1This relation relates math instructors and the sandwiches they enjoy.

24


Example 4. Determine whether the equation defines y as a function of x.

x2 + y = 1


x + y2 = 1


x2 + y2 = 1


x3 + y3 = 1

25


Example 8. Evaluate the function for the given values.

f(x) = x2 + 2x + 1

f(4) =

f(−x) =

f(x + h) =


f(x) = x2 − x− 6

f(−3) =

f(−x) =

f(x + h) =


f(x) = x3 − 3x2 + 3x− 1

f(2) =

f(−x) =

26


Extremum, Symmetry, Piecewise Functions, and the Difference Quotient

Increasing Functions, Decreasing Functions, Constant Functions

Let f be a function and (a, b) be some interval in the domain of f . The function is called

• increasing over (a, b) if f(x) < f(y) for every x < y,

• decreasing over (a, b) if f(x) > f(y) for every x < y, and

• constant over (a, b) if f(x) = f(y) for every x and y

(where a < x < y < b).

Example 1. Determine over which intervals the function f is increasing, decreasing, or constant.

Increasing:

Decreasing:

Constant:

Relative Maximum:

Relative Minimum:

Domain:

Range:

Zeros of the function:

27



Increasing:

Decreasing:

Constant:

Relative Maximum:

Relative Minimum:

Domain:

Range:


Increasing:

Decreasing:

Constant:

28


Even and Odd Functions

A function f is called even iff(−x) = f(x).

A function f is called odd iff(−x) = −f(x).

Example 4. Determine if f is even, odd, or neither.

f(x) = x2 − 4

Example 5. Determine if g is even, odd, or neither.

g(x) = x3 − 2x

Example 6. Determine if h is even, odd, or neither.

h(x) = (x− 2)2

29


Example 7. Evaluate the piecewise function.

f(x) =

2x + 8 if x ≤ −2

x2 if − 2 < x ≤ 1

1 if x > 1

f(−3) =

f(−1) =

f(2) =

f(4) =

Example 8. Evaluate the piecewise function.

f(x) =

{x if x ≥ 0

−x if x < 0

f(−2) =

f(−1) =

f(1) =

f(2) =

Example 9. Graph the piecewise function.

f(x) =

{x + 2 if x ≤ 0

1 if x > 0

x

y

−3 −2 −1 1 2 3

−1

1

2

3

30



f(x) =

{x if x ≥ 0

−x if x < 0

x

y

−3 −2 −1 1 2 3

−1

1

2

3


f(x) =

2x + 8 if x ≤ −2

x2 if − 2 < x ≤ 1

1 if x > 1

x

y

−3 −2 −1 1 2 3

−1

1

2

3

4

31


Difference Quotient

For a function f(x), the difference quotient is

f(x + h)− f(x)

h, h 6= 0.

Example 12. Find the difference quotient of the given function.

f(x) = 2x + 3


f(x) = 5x− 6

32



f(x) = x2 + 1


f(x) = x2 − 4x

33


Graphing Functions

The function f(x) = x2 is called the square function.

x f(x)

−2 4−1 10 01 12 4

The function f(x) = x3 is called the cube function.

x f(x)

−2 −8−1 −10 01 12 8

The function f(x) =√x is called the square root function.

x f(x)

0 01 14 29 3

34


The function f(x) = 3√x is called the cube root function.

x f(x)

−8 −2−1 −10 01 18 2

The function f(x) = |x| is called the absolute value function.

x f(x)

−2 2−1 10 01 12 2

Transformations of f(x)

f(x) + c vertical shift up c unitsf(x)− c vertical shift down c unitsf(x + c) horizontal shift left c unitsf(x− c) horizontal shift right c units−f(x) reflection over the x-axisf(−x) reflection over the y-axiscf(x) vertical stretch or compression by a factor of cf(cx) horizontal compression or stretch by a factor of c

35


Example 1. Graph g(x) = x2 + 1.

Let f(x) = x2. Note g(x) =f(x) + 1.

x f(x) g(x)

−2 4 5−1 1 20 0 11 1 22 4 5

Example 2. Graph g(x) = (x− 2)2.

Let f(x) = x2. Note g(x) =f(x− 2).

x− 2 f(x− 2) x g(x)

−2 4 0 4−1 1 1 10 0 2 01 1 3 12 4 4 4

Example 3. Graph h(x) = (x + 4)2 + 1.

Let f(x) = x2 and g(x) = (x + 4)2.

Note h(x) =g(x) + 1 =f(x + 4) + 1.

x + 4 f(x + 4) x g(x) x h(x)

−2 4 −6 4 −6 5−1 1 −5 1 −5 20 0 −4 0 −4 11 1 −3 1 −3 22 4 −2 4 −2 5

36


Example 4. Graph k(x) = −(x− 3)2 − 1.

Let f(x) = x2, g(x) = (x− 3)2, and h(x) = −(x− 3)2.

Note k(x) =h(x)− 1 =−g(x)− 1 =−f(x− 3)− 1.

x− 3 f(x− 3) x g(x) x h(x) x k(x)

−2 4 1 4 1 −4 1 −5−1 1 2 1 2 −1 2 −20 0 3 0 3 0 3 −11 1 4 1 4 −1 4 −22 4 5 4 5 −4 5 −5

Example 5. Graph g(x) =√x− 2.

Let f(x) =√x. Note g(x) =f(x)− 2.

x f(x) g(x)

0 0 −21 1 −14 2 09 3 1

37


Example 6. Graph g(x) =√x− 2.

Let f(x) =√x. Note g(x) =f(x− 2).

x− 2 f(x) x g(x)

0 0 2 01 1 3 14 2 6 29 3 11 3

Example 7. Graph k(x) = −√x + 1 + 2.

Let f(x) =√x, g(x) =

√x + 1, and h(x) = −

√x + 1.

Note k(x) =h(x) + 2 =−g(x) + 2 =−f(x + 1) + 2.

x + 1 f(x + 1) x g(x) x h(x) x k(x)

0 0 −1 0 −1 0 −1 21 1 0 1 0 −1 0 14 2 3 2 3 −2 3 09 3 8 3 8 −3 8 −1

38


Example 8. Graph h(x) = (x− 5)3 − 2.

Let f(x) = x3 and g(x) = (x− 5)3.

Note h(x) =g(x)− 2 =f(x− 5)− 2.

x− 5 f(x− 5) x g(x) x h(x)

−2 −8 3 −8 3 −10−1 −1 4 −1 4 −30 0 5 0 5 −21 1 6 1 6 −12 8 7 8 7 6

Example 9. Graph k(x) = −|x + 2|+ 1.

Let f(x) = |x|, g(x) = |x + 2|, and h(x) = −|x + 2|.

Note k(x) =h(x) + 1 =−g(x) + 1 =−f(x + 2) + 1.

x + 2 f(x− 3) x g(x) x h(x) x k(x)

−2 2 −4 2 −4 −2 −4 −1−1 1 −3 1 −3 −1 −3 00 0 −2 0 −2 0 −2 11 1 −1 1 −1 −1 −1 02 2 0 2 0 −2 0 −1

39


Example 10. The graph of the function f is given below.

(a) Graph g(x) = f(x)− 2.

(b) Graph h(x) = f(x + 2).

(c) Graph k(x) = −f(x− 1) + 2.

40


Unit 2

Quadratic Functions

Vertex

The vertex of a parabola is the point where the parabola achieves its minimum or maximum value.

Example 1. Graph f(x) = x2.

Vertex:

Example 2. Graph f(x) = (x + 4)2 − 1.

Vertex:

Example 3. Graph f(x) = 2(x− 5)2 + 3.

Vertex:

41


Example 4. Graph f(x) = −3(x− 3)2 − 1.

Vertex:

Standard and General Form of a Parabola

A quadratic function is said to be in standard form if it is written as

f(x) = a(x− h)2 + k.

A quadratic function is said to be in general form if it is written as

f(x) = ax2 + bx + c.

Example 5. Graph f(x) = x2 + 8x + 15.

Vertex:

42


Example 6. Graph f(x) = x2 + 6x + 7.

Vertex:

Example 7. Graph f(x) = x2 − 8x + 19.

Vertex:

43


Example 8. Graph f(x) = 2x2 − 4x− 3.

Vertex:

Example 9. Graph f(x) = x2 − 5x + 1.

Vertex:

44


Example 10. Write the quadratic function f(x) = ax2 + bx + c in standard form.

Vertex:

45


Example 11. Evaluate the quadratic function f(x) = ax2 + bx + c for x = − b2a .

Vertex Formula

For any quadratic function f(x) = ax2 + bx + c, the vertex is located at(− b

2a, f

(− b

2a

)).

Example 12. Graph f(x) = 3x2 − 6x + 2.

Example 13. Graph f(x) = −2x2 − 7x + 5.

46


Polynomial Functions

Polynomial Function

A polynomial function is a function of the form

f(x) = anxn + an−1x

n−1 + ... + a2x2 + a1x + a0,

where n is a nonnegative integer and each ai is a real number. Assuming an 6= 0, the degree of thepolynomial function is n and an is called the leading coefficient.

Example 1. Graph the following power functions.

a. f(x) = x

b. f(x) = x2

c. f(x) = x3

d. f(x) = x4

e. f(x) = x5

f. f(x) = x6

g. f(x) = x7

End Behavior

The end behavior of a function is the value f(x) approaches as x approaches −∞ or as x ap-proaches ∞.

Example 2. Identify the end behavior of each of the power functions in Example 1.

Power Function x −→ −∞ x −→∞f(x) = x f(x) −→ f(x) −→f(x) = x2 f(x) −→ f(x) −→f(x) = x3 f(x) −→ f(x) −→f(x) = x4 f(x) −→ f(x) −→f(x) = x5 f(x) −→ f(x) −→f(x) = x6 f(x) −→ f(x) −→f(x) = x7 f(x) −→ f(x) −→

47


End Behavior of Any Polynomial Function

The end behavior of any polynomial function is the same as the end-behavior of its highest degreeterm.

Zero of a Function

If f(c) = 0, then c is called a zero of the function.

If c is a zero of a function, then (c, 0) is an x-intercept on the graph of the function.

Example 3. Sketch a graph of f(x) = (x + 4)(x + 1)(x− 2).

Example 4. Sketch a graph of f(x) = (x + 4)(x + 1)2(x− 2).

48


Example 5. Sketch a graph of f(x) = (x + 4)2(x + 1)3(x− 2).

Multiplicity of a Zero

If (x − c)n is a factor of f(x), but (x − c)n+1 is not a factor of f(x), then c is a zero of multiplic-ity n.

If c is a zero of multiplicity n, then:

• if n is odd, the graph crosses the x-axis,

• if n is even, the graph touches the x-axis, but does not cross.

The Intermediate Value Theorem

If f(x) is a polynomial function and a and b are real numbers with a < b, then if either

• f(a) < 0 < f(b), or

• f(b) < 0 < f(a),

then there exists a real number c such that a < c < b and f(c) = 0.

Example 6. Use the Intermediate Value Theorem to verify f(x) = x3 + x + 1 has a zero on the closedinterval [−1, 0].

49


The Division Algorithm

The Division Algorithm

Let p(x) be a polynomial of degree m and let d(x) be a nonzero polynomial of degree n wherem ≥ n. Then there exists unique polynomials q(x) and r(x) such that

p(x) = d(x) · q(x) + r(x)

where the degree of q(x) is m− n and the degree of r(x) is less than n. The polynomial d(x) is called thedivisor, q(x) is called the quotient, and r(x) is called the remainder.

Example 1. Use long division to dividex2 − 3x− 6

x + 4. State the quotient, q(x), and remainder, r(x),

guaranteed by the Division Algorithm.

Example 2. Use long division to dividex4 − 4x3 + 6x2 − 4x + 1

x− 1. State the quotient, q(x), and remainder,

r(x), guaranteed by the Division Algorithm.

50


Example 3. Use long division to dividex3 + x + 1

x + 1. State the quotient, q(x), and remainder, r(x), guar-

anteed by the Division Algorithm.

Example 4. Use synthetic division to dividex3 − 3x2 − 10x + 24

x− 2. State the quotient, q(x), and remain-

der, r(x), guaranteed by the Division Algorithm.

Example 5. Use synthetic division to dividex2 − 10x + 24

x + 5. State the quotient, q(x), and remainder,

r(x), guaranteed by the Division Algorithm.

51


Example 6. Use synthetic division to dividex3 − 7x + 12

x− 3. State the quotient, q(x), and remainder, r(x),

guaranteed by the Division Algorithm.

The Remainder Theorem

Let p(x) be a polynomial. Then p(c) = r(x) where r(x) is the remainder guarenteed from the di-vision algorithm with d(x) = x− c.

Example 7. Evaluate f(x) = x4 − 3x3 + 5x2 − 7x + 8 for f(2) using the remainder theorem.

Example 8. Evaluate f(x) = x5 + 4x3 − 9x + 2 for f(−1) using the remainder theorem.

52


The Fundamental Theorem of Algebra

The Factor Theorem

Let p(x) and d(x) be polynomials. If r(x) = 0 by the division algorithm, then d(x) is a factor ofp(x).

Example 1. Use the remainder theorem to verify that −3 is a zero of f(x) = x3 − 3x2 − 10x + 24. Thenfind all other zeros.

Example 2. Use the remainder theorem to verify that 7 is a zero of f(x) = x3− 5x2− 13x− 7. Then findall other zeros.

53


The Rational Zeros Theorem

Let p(x) be a polynomial function with integer coefficients:

p(x) = anxn + an−1x

n−1 + ... + a1x + a0.

Then any rational zero of the polynomial will be of the form

± factor of a0factor of an

leading coefficient an and constant term a0.

Example 3. List all possible rational zeros for f(x) = x3 − 3x2 − 10x + 24 given by the Rational ZerosTheorem.

Example 4. List all possible rational zeros for f(x) = 2x3 + 3x2 − 32x + 15 given by the Rational ZerosTheorem.

Descartes’ Rule of Signs

Let p(x) be a polynomial function.

The number of positive real zeros is equal to or less than by an even number the number of signchanges of p(x).

The number of negative real zeros is equal to or less than by an even number the number of signchanges of p(−x).

54


The Fundamental Theorem of Algebra

Let p(x) be a polynomial function of degree n. Then p(x) has n complex zeros, including multi-plicities.

Example 5. Find all zeros of the function f(x) = x3 − 4x2 + x + 6.

Example 6. Find all zeros of the function f(x) = x3 + 7x2 + 16x + 12.

55


Example 7. Find all zeros of the function f(x) = x4 − 4x3 − 19x2 + 46x− 24.

Example 8. Find all zeros of the function f(x) = x4 − x3 − 2x2 − 4x− 24.

56


Complex Conjugate Theorem

Let p(x) be a polynomial with real coefficients. If a + bi is a zero of the polynomial, then itscomplex conjugate a− bi is also a zero of the polynomial.

Example 9. Find a third degree polynomial f(x) with zeros of i and 3 such that f(0) = −3.

Example 10. Find a third degree polynomial f(x) with zeros of 1 + i and −1 such that f(1) = 2.

57


Rational Functions

Example 1. State the domain of the rational function.

f(x) =x− 1

x2 − x− 6

Example 2. Graph the rational function.

f(x) =1

x


f(x) =1

x− 3+ 2

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Vertical Asymptotes

Let r(x) =n(x)

d(x)be a simplified rational function. If c is a zero of d(x), then x = c is a vertical

asymptote.

Horizontal Asymptotes

Let r(x) =n(x)

d(x)be a rational function.

1. If the degree of the denominator, d(x), is greater than the degree of the numerator, n(x), then theline y = 0 is the horizontal asymptote.

2. If the degree of the denominator, d(x), is equal to the degree of the numerator, n(x), then the liney = ab is the horizontal asymptote, where a is the leading coefficient of n(x) and b is the leadingcoefficent of d(x).

3. If the degree of the denominator, d(x), is less than the degree of the numerator, n(x), then there isno horizontal asymptote.

Holes

Let r(x) =f(x) · n(x)

f(x) · d(x)be a rational function. If c is a zero of f(x), then there is a hole at

(c,n(c)

d(c)

).

Example 4. Find any vertical or horizontal asymptotes. Identify any holes in the graph.

f(x) =3x

x2 − 9

59



f(x) =3x2

x2 − 9


f(x) =3x3

x2 − 9


f(x) =3x + 9

x2 − 9

60



f(x) =1

x2 + x− 6


f(x) =2x

x2 − 4


f(x) =1

x2 + 1

61


Polynomial and Rational Inequalities

Example 1. Solve.x2 − 7x + 12 = 0

Solution:

Example 2. Solve.x2 − 7x + 12 > 0

Solution:

Example 3. Solve.x2 + x ≤ 20

Solution:

62


Example 4. Solve.4x2 ≥ 4x + 3

Solution:

Example 5. Solve.x− 3

x + 4≥ 0

Solution:

63


Example 6. Solve.2x2 − 5x + 3

2− x≥ 0

Solution:

Example 7. Solve.x

x + 4≥ 2

Solution:

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Unit 3

Operations on Functions

Basic Operations on Functions

Let f(x) and g(x) be functions. The following basic operations of addition, subtraction, multipli-cation, and division may be performed on the functions as follows:

• (f + g)(x) = f(x) + g(x)

• (f − g)(x) = f(x)− g(x)

• (f · g)(x) = f(x) · g(x)

•(f

g

)(x) =

f(x)

g(x)

If the domain of f(x) is A and the domain of g(x) is B, then the domain of f + g, f − g, and f · g is A∩B.The domain of f/g is A ∩B restricted for any x values such that g(x) = 0.

Example 1. Let f(x) = 3x− 2 and g(x) = x + 7. Find f + g, f − g, f · g, and f/g. State the domain ofeach function.

65


Composition of Functions

Let f(x) and g(x) be functions. The composition of f and g, denoted f ◦ g, is given by

(f ◦ g)(x) = f(g(x)).

Example 2. Let f(x) = 3x− 2 and g(x) = x+ 7. Find f ◦ g and g ◦ f . State the domain of each function.

66


Example 3. Let f(x) =1

x + 7and g(x) =

x

x + 3. Find f ◦ g and g ◦f . State the domain of each function.

67


Inverse Functions

Inverse Functions

Two functions f and g are called inverse functions if

(f ◦ g)(x) = (g ◦ f)(x) = x.

Example 1. Verify that f and g are inverse functions. Graph both f and g.

f(x) = 3x + 4 g(x) =x− 4

3

Example 2. Verify that f and g are inverse functions. Graph both f and g.

f(x) = x3 + 2 g(x) = 3√x− 2

68


Recall the definition of a function:

One-to-one Function

A function f is called one-to-one if each element in the range corresponds to exactly one elementin the domain. If a function is one-to-one, then it has an inverse function.

Example 3. Which of the following functions are one-to-one?

69


Example 4. Determine the inverse of the one-to-one function. State the domain and range of f and f−1.

f(x) =1

2x− 3


f(x) = x2 − 3, x ≥ 0

70



f(x) =√x− 4


f(x) = 3√x + 2

71


Exponential Functions

Exponential Function

The functionf(x) = bx,

where b > 0 and b 6= 1, is called an exponential function.

Example 1. Graph f(x) = 2x. State its domain and range.

Example 2. Graph f(x) = 2x + 3. State its domain and range.

Example 3. Graph f(x) = 2x+3. State its domain and range.

72


Example 4. Graph f(x) = 2−x. State its domain and range.

Example 5. Graph f(x) = 3x+2 − 4. State its domain and range.

Natural Base

Consider the expression(1 + 1

n

)nfor various values of n. See the table below.

n(1 + 1

n

)n1 210 2.59374100 2.70481

1,000 2.7169210,000 2.71815100,000 2.71827

As n gets bigger, the expression(1 + 1

n

)ngets bigger as well, but this sequence has an upper bound. This

particular upper bound is called the natural base, e.

e = 2.71828...

Example 6. Graph f(x) = ex. State its domain and range.

73


Example 7. Graph f(x) = e−x. State its domain and range.

Example 8. Graph f(x) = −3ex + 1. State its domain and range.

Example 9. Graph f(x) = e2x − 3. State its domain and range.

74


Logarithmic Functions

Logarithm

If by = x, then logb x = y. The expression logb x is read “the logarithm base b of x” or “log baseb of x.”

Example 1. Write the following exponential equations as logarithmic equations.

(a) 24 = 16

(b) 53 = 125

(c) 8112 = 9

(d) 4−3 =1

64

Example 2. Write the following logarithmic equations as exponential equations.

(a) log10 1000 = 3

(b) log3 243 = 5

(c) log27 3 =1

3

(d) −3 = log21

8

Example 3. Evaluate the following logarithms.

(a) log5 25

(a) log10 10000

(a) log3 1

(a) log2 64

(a) log361

6

75


Example 4. Find the inverse function of f(x) = 2x.

Example 5. Graph the function f(x) = 2x and g(x) = log2 x.

76


Example 6. Graph the function f(x) = log3 x. State the domain and range.

Natural and Common Logarithms

The natural logaritm of x is denoted lnx and lnx = loge x.

The common logaritm of x is denoted log x and log x = log10 x.

Example 7. Graph the function f(x) = log x. State the domain and range.

Example 8. Graph the function f(x) = lnx. State the domain and range.

77


Example 9. Graph the function f(x) = log2(x− 3). State the domain and range.

Example 10. Graph the function f(x) = log2 x− 3. State the domain and range.

78


Properties of Logarithms

Basic Properties of Logarithms

1. logb 1 = 0

2. logb b = 1

3. logb bx = x

4. blogb x = x

Example 1. Evaluate the following expressions.

(a) log8 1

(b) log4 4

(c) log2 27

(d) 3log3 4

Example 2. Evaluate the following expressions.

(a) ln 1

(b) log 10

(c) ln e3

(d) eln(2x)

79


Product Rule for Logarithms

logb(M ·N) = logbM + logbN

Proof.

Quotient Rule for Logarithms

logb

(M

N

)= logbM − logbN

Proof.

80


Power Rule for Logarithms

logb(MN

)= N logbM

Proof.

Example 3. Expand the logarithmic expression.

log3

(x2y

9z3

)

Example 4. Condense the logarithmic expression.

lnx + 5 ln y − 3 ln z

81


Change of Base Formula

logbM =logaM

loga b

Proof.

Example 5. Approximate the logarithm.log4 50

82


Exponential and Logarithmic Equations

One-to-one Property for Exponential Functions

If bx = by, then x = y.

Example 1. Solve the equation.34x+1 = 81

Example 2. Solve the equation.43x−1 = 8x+5

83


One-to-one Property for Logarithmic Functions

If x = y and x > 0, then logb x = logb y.

Example 3. Solve the equation.7x−3 = 21

Example 4. Solve the equation.42x−3 = 53x+4

84


Example 5. Solve the equation.e2x+5 = 18

Example 6. Solve the equation.e2x − ex − 6 = 0

85


Example 7. Solve the equation.log3(x + 3) = 2

Example 8. Solve the equation.ln(x + 7) = 4

86


Example 9. Solve the equation.log2(x− 5) + log2(x + 2) = 3

Second One-to-one Property for Logarithmic Functions

If logb x = logb y and x > 0 and y > 0, then x = y.

Example 10. Solve the equation.ln(x + 3)− lnx = ln 7

87


Unit 4

Circles

Distance Formula

The distance between any two points (x1, y1) and (x2, y2) is given by

d =√

(x2 − x1)2 + (y2 − y1)2.

y1

y2

x1 x2

(x2, y2)

(x1, y1) x2 − x1

y2 − y1d

Example 1. Find the distance between the points (−3, 4) and (3,−1).

88


Midpoint Formula

The point halfway between two points (x1, y1) and (x2, y2) is called the midpoint and is givenby (

x1 + x22

,y1 + y2

2

).

y1

y2

x1 x2

(x2, y2)

(x1, y1)

(x1 + x2

2,y1 + y2

2

)

Example 2. Locate the midpoint between (−3, 4) and (3,−1).

89


Example 3. Find the distance between (h, k) and any point (x, y).

Circles

A circle is the set of all points a fixed distance, called the radius, from a fixed point, called thecenter.

Centered at (h, k) with radius r

(x− h)2 + (y − k)2 = r2

Example 4. Graph x2 + y2 = 1.

90


Example 5. Graph (x− 3)2 + (y + 1)2 = 9.

Example 6. Graph x2 + 8x + y2 − 4y − 16 = 0.

91


Ellipses

Ellipses Centered at (0, 0)

An ellipse is the set of all points a fixed distance from two fixed point, called the foci.

For both of the following equations, a > b and c2 = a2 − b2.

Horizontal Major Axis

x2

a2+

y2

b2= 1

(0, 0) (a, 0)(−a, 0)

(0, b)

(0,−b)

The foci are located at (c, 0) and (−c, 0).

Vertical Major Axis

x2

b2+

y2

a2= 1

(b, 0)(−b, 0)

(0, a)

(0,−a)

The foci are located at (0, c) and (0,−c).

92


Example 1. Graphx2

25+

y2

9= 1.

Example 2. Graphx2

4+

y2

16= 1.

Example 3. Graph 9x2 + 16y2 = 144.

Example 4. Write the equation of the ellipse that has vertices at (0, 6) and (0,−6) and foci at (0, 5) and(0,−5).

93


Ellipses Centered at (h, k)

Horizontal Major Axis

(x− h)2

a2+

(y − k)2

b2= 1

The vertices are located at (h + a, k) and (h− a, k).

The covertices are located (h, k + b) and (h, k − b).

The foci are located at (h + c, k) and (h− c, k).

Vertical Major Axis

(x− h)2

b2+

(y − k)2

a2= 1

The vertices are located at (h, k + a) and (h, k − a).

The covertices are located (h + b, k) and (h− b, k).

The foci are located at (h, k + c) and (h, k − c).

Example 5. Graph(x + 4)2

20+

(y − 2)2

36= 1.

Example 6. Graph x2 + 9y2 = 9.

94


Example 7. Graph 4x2 − 8x + 9y2 + 90y + 193 = 0.

Example 8. Graph 4x2 + 24x + y2 − 10y + 57 = 0.

95


Hyperbolas

Hyperbolas Centered at (0, 0)

A hyberbola is the set of all points a fixed distance, when you subtract, from two fixed point, called the foci.

For both of the following equations, c2 = a2 + b2.

Horizontal Transverse Axis

x2

a2− y2

b2= 1

(a, 0)(−a, 0)

(0, b)

(0,−b)

(c, 0)(−c, 0)

The vertices are located at (a, 0) and (−a, 0).The foci are located at (c, 0) and (−c, 0).

Vertical Transverse Axis

y2

a2− x2

b2= 1

(b, 0)(−b, 0)

(0, a)

(0,−a)

(0, c)

(0,−c)

The vertices are located at (0, a) and (0,−a).The foci are located at (0, c) and (0,−c).

96


Example 1.

Example 2.

97


Example 3.

Example 4.

98


Example 5.

Example 6.

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College Algebra Notes - Metropolitan Community College · PDF filex+ 3 + 2 x+ 2 x+ 3 Example 16. Simplify the complex rational expression. State any domain restrictions. 1 + 6 x +

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