Chapter 1 – Functions and Their Graphs
Section 1Functions
Introduction to Functions
Definition – A function f from a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is the domain of the function f, and the set B contains the range
Characteristics of a Function
1. Each element in A must be matched with an element in B.
2. Some elements in B may not be matched with any element in A.
3. Two or more elements in A may be matched with the same element in B.
4. An element in A (domain) cannot be match with two different elements in B
Example
A = {1,2,3,4,5,6} and B = {9,10,12,13,15}
Is the set of ordered pairs a function?
{(1,9), (2,13), (3,15), (4,15), (5,12), (6,10)}
Vertical Line Test
Use the vertical line test to determine graphically when you have a function. If you can draw a vertical line and it does not pass through more than one point on the graph, then the graph depicts a function.
Function NotationThe variable f is usually used to depict a function.
It is only notation, and f(x) simply replaces y in your typical equations and is read f of x.
Therefore y = f(x)
That means if y = 2x +4 then an equivalent equation using function notation is
f(x) = 2x + 4 Nothing changes, it’s just another use of
symbols.
ExampleEvaluate the function when x = -1, 0, and 1
f(x) = { x2 +1, x< 0 { x -1, x ≥ 0
f(-1) = (-1)2 +1 = 2f(0) = 0 -1 = -1f(1) = 1 – 1) = 0
f(x) = 1 – x2 then
f(1) = 1 – (1) 2 = 0f(2) = 1 – (2) 2 = -3f(0) = 1 – (0) 2 = 1
Domain of a Function
The domain of a function is the set of all real numbers for which the expression is defined.
EXAMPLE
f(x) = 1/(x2 -4)
The domain is the set of real numbers excluding ± 2.
Section 2
Analyzing Graphs of Functions
Graph of a FunctionThe graph of a function f is the collection of
ordered pairs (x,f(x)) such that x is in the domain of f.
Domain – is the set of all x values
Range – is the set of all f(x) values
Zeros of a FunctionThe zeros of a function f of x are the x-values
for which f(x) = 0
EXAMPLEFind the zeros of f(x) = 3x2 +x - 10
3x2 +x – 10 = 0 - Factor and solve for x
Increasing and Decreasing Functions
1. A function f is increasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2)
2. A function f is decreasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2)
3. A function f is constant on an interval if, for any x1 and x2 in the interval, f(x1) = f(x2)
Example1. Graph f(x) = x3
2. Graph f(x) = x3 – 3x
3. Graph f(x) = {x +1, x < 0 {1, 0 ≤ x ≤ 2 { -x + 3, x >2
Definition of Relative Minimum and Relative
Maximum A function value f(a) is called a relative
minimum of f if there exist an interval (x1, x2) that contains a such thatx1 < x < x2 implies f(a) ≤ f(x)
A function value f(a) is called a relative maximum of f if there exist an interval (x1, x2) that contains a such thatx1 < x < x2 implies f(a) ≥ f(x)
Example1. Graph f(x) = 3x2 – 4x -2 using a calculator
to estimate the relative minimum or relative maximum
2. Graph f(x) = -3x2 + 4x + 2 using a calculator to estimate the relative minimum or relative maximum
Types of Functions Linear Functions:
f(x) = mx + b
Step Functions:f(x) = [[ x ]] = greatest integer less than or equal to x
Piecewise-Defined Functions:f(x) = {2x +3, x ≤ 1 {- x + 4, x > 1
Even and Odd Functions A function y = f(x) is even if, for each x in
the domain of f, f(-x) = f(x) --- symmetric to y-axis
A function y = f(x) is odd if, for each x in the domain of f, f(-x) = - f(x) --- symmetric to origin
ExampleDetermine whether each function is even,
odd, or neither
1. g(x) = x3 –x
2. h(x) = x2 + 1
Section 3
Shifting, Reflecting, and Stretching Graphs
Summary of Graphs of Common Functions
f(x) = c f(x) = x f(x) = |x| f(x) = x f(x) = x2
f(x) = x3
Shifting Graphs Transforms graphs by shifting
upward, downward, left or right with basic graph the same.
EXAMPLE
h(x) = x2 + 2 shifts the graph upward two units
Vertical Shiftsh(x) = f(x) + c for c > 0 Vertical shift c units upward
f(x) = f(x) – c for c > 0 Vertical shift c units downward
Horizontal Shiftsh(x) = f(x – c) for c > 0 horizontal shift c units right
f(x) = f(x + c) for c > 0 horizontal shift c units left
Reflecting in the Coordinate Axes
Reflections in the x-axis: h(x) = - f(x)
Reflections in the y-axis: h(x) = f(-x)
Reflecting GraphsTransforms graphs by
creating a mirror image
EXAMPLE
If h(x) = x2 then g(x) = - x2 is the reflection
Nonrigid Transformation Transformations that cause a
distortion – a change in the shape of the original graph
If h(x) = |x| then g(x) = 3|x| is a vertical stretch of h(x)
but p(x) = ⅓|x| would be a vertical shrink
Section 4Combinations of Functions
Arithmetic Combinations of Functions
(f +g)(x) = f(x) + g(x) sum
(f -g)(x) = f(x) - g(x) difference
(fg)(x) = f(x) · g(x) product
(f/g)(x) = f(x)/g(x), g(x) ≠ 0 quotient
Examplesf(x) = 2x + 1 and g(x) = x2 + 2x – 1
Find: (f +g)(x) = f(x) + g(x)
= x2 + 4x Find: (fg)(x) = f(x)· g(x) =2 x3 +5x2 - 1
Composition of FunctionsThe composition of the function f with the
function g is (f ◦ g)(x) = f(g(x))
The domain of f ◦ g is the set of all x in the domain of g such that g(x) is in the domain of f
Examplesf(x) = x + 2 and g(x) = 4 – x2
Find: (f ◦ g)(x) = f(g(x))
= f(4 – x2) Simplify = 4 – x2+2
Find: (g ◦ f)(x) = g(f(x)) = g(x + 2) Simplify = 4 – (x +2)2
Examplesf(x) = x2 - 9 and g(x) = (9 - x2)½
Find: domain of (f ◦ g)
Remember the domain of (f ◦ g) is the set of all x in the domain of g
Find domain of g(x):
Section 5Inverse of Functions
Inverse FunctionsLet f and g be two functions such that
f(g(x)) = x for every x in the domain of g and;
g(f(x)) = x for every x in the domain of f
Under these conditions, the function g is the inverse function of the function f
Inverse FunctionsThe inverse function is formed by
interchanging the first and second coordinates of each of the ordered pairs and the inverse is denoted by f -1
Again, this is simply notation!
The domain of f must be equal to the range of f -1 , and the range of f must be equal to the domain of f-1
ExampleFind the inverse function of f(x) = 2x - 3 Replace f(x) with y and solve for x y = 2x -3 x = (y+3)/2
Now interchange x and y and you have f -1
y = (x+3)/2
Guidelines for Finding an Inverse Function
1. Use the Horizontal Line Test to decide whether f has an inverse function
2. In the equation for f(x), replace f(x) by y
3. Interchange the roles of x and y, and solve for y.
4. Replace y by f-1(x) in the new equation
5. Verify that f and f-1 are inverse functions of each other by showing that the domain of f is equal to the range of f -1 and the range of f is equal to the domain of f -1
Section 6Mathematical Modeling
Direct VariationThe following statements are equivalent.
1. y varies directly as x.2. y is directly proportional to x3. y = kx for some nonzero constant k
EXAMPLE
D = rt F= ma
Inverse VariationThe following statements are equivalent.
1. y varies inversely as x.2. y is inversely proportional to x3. y = k/x for some nonzero constant k
EXAMPLE
V = kT/P
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