NAME DATE PERIOD 1-5 Study Guide and · PDF fileLesson 1-5 NAME DATE PERIOD ... Lesson 1-6 NAME DATE PERIOD ... Given functions f and g, the composite function f
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Parent Functions A parent function is the simplest of the functions in a family.
Parent Function Form Notes
constant function f(x) = c graph is a horizontal line
identity function f(x) = x points on graph have coordinates (a, a)
quadratic function f(x) = x2 graph is U-shaped
cubic function f(x) = x3 graph is symmetric about the origin
square root function f(x) = √ �
x graph is in first quadrant
reciprocal function f(x) = 1 − x graph has two branches
absolute value function f(x) = | x | graph is V-shaped
greatest integer function f(x) = �x�
defined as the greatest integer less than or equal to x; type of step function
Describe the following characteristics of the graph of the parent function f(x) = x3 : domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing.
The graph confirms that D = {x | x ∈ �} and R = {y | y ∈ �}.
The graph intersects the origin, so the x-intercept is 0 and the y-intercept is 0.
It is symmetric about the origin and it is an odd function:
f(-x) = -f(x).
The graph is continuous because it can be traced without lifting the pencil off the paper.
As x decreases, y approaches negative infinity, and as x increases, y approaches positive infinity.
lim x → -∞
f(x) = -∞ and lim x → ∞
f(x) = ∞
The graph is always increasing, so it is increasing for (-∞, ∞).
Exercise
Describe the following characteristics of the graph of the parent function f(x) = x2 : domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing.
Study Guide and InterventionParent Functions and Transformations
Transformations of Parent Functions Parent functions can be transformed to create other members in a family of graphs.
Translations
g(x) = f(x) + k is the graph of f(x) translated…
…k units up when k > 0.
…k units down when k < 0.
g(x) = f(x - h) is the graph of f(x) translated…
…h units right when h > 0.
…h units left when h < 0.
Reflections
g(x) = -f(x) is the graph of f(x)… …reflected in the x-axis.
g(x) = f(-x) is the graph of f(x)… …reflected in the y-axis.
Dilations
g(x) = a � f(x) is the graph of f(x)…
…expanded vertically if a > 1.
…compressed vertically if 0 < a < 1.
g(x) = f(ax) is the graph of f(x)…
…compressed horizontally if a > 1.
…expanded horizontally if 0 < a < 1.
Identify the parent function f(x) of g(x) = √��-x - 1, and describe how the graphs of g(x) and f(x) are related. Then graph f(x) and g(x) on the same axes.
The graph of g(x) is the graph of the square root function f(x) = √�x reflected in the y-axis and then translated one unit down.
ExercisesIdentify the parent function f(x) of g(x), and describe how the graphs of g(x) and f(x) are related. Then graph f(x) and g(x) on the same axes.
2. Use the graph of f(x) = ⎪x⎥ to graph g(x) = -|2x|.
y
x
3. Describe how the graph of f(x) = x2 and g(x) are related. Then write an equation for g(x).
4. Identify the parent function f(x) of g(x) = 2|x + 2| - 3. Describe how the graphs of g(x) and f(x) are related. Then graph f(x) and g(x) on the same axes.
5. Graph f(x) =
y
x
6. Use the graph of f(x) = x3 to graph g(x) = ⎪(x + 1)3
1. AREA The width w of a rectangular plot of land with fixed area A is modeled by the function w(�) = A
−
�
, where � is the length.
a. If the area is 1000 square feet, describe the transformations of the parent function f(x) = 1
−x used to graph w(x).
b. Describe a function of the length that could be used to find a minimum perimeter for a given area
c. Is the function you found in part b a transformation of f(x)? Explain.
d. Find the minimum perimeter for an area of 1000 square feet.
2. GOLF The path of the flight of a golf ball
can be modeled by h(x) = -
1−
10x2
+ 2x, where h(x) is the distance above the ground in yards and x is the horizontal distance from the tee in yards.
a. Describe the transformation of the parent function f(x) = x2 used to graph h(x).
b. Suppose the same shot was made from a tee located 10 yards behind the original tee. Rewrite h(x) to reflect this change.
3. TAXES Graph the tax rates for the different incomes by using a step function.
4. HORIZON The function f(x) = √��1.5x can be used to approximate the distance to the apparent horizon, or how far a person can see on a clear day, where f(x) is the distance in miles and x is the person’s elevation in feet.
a. How does the graph of f(x) compare to the graph of its parent function?
b. The function g(x) = 1.2 √�x is also
used to approximate the distance to the horizon. How does the graph of g(x) compare to the graph of its parent function?
Word Problem PracticeParent Functions and Transformations
Operations with Functions Two functions can be added, subtracted, multiplied, or divided to form a new function. For the new function, the domain consists of the intersection of the domains of the two functions, excluding values that make a denominator equal to zero.
Given f(x) = x2 - x - 6 and g(x) = x + 2, find each function and its domain. a. (f + g)(x)
(f + g)x = f(x) + g(x) = x2 - x - 6 + x + 2
= x2 - 4
The domains of f and g are both (-∞, ∞), so the domain of (f + g) is (-∞, ∞).
b. ( f − g ) (x)
( f − g ) x =
f(x) −
g(x)
= x2 - x - 6 −
x + 2
= (x - 3)(x + 2) −
x + 2 = x - 3
The domains of f and g are both (-∞, ∞), but x = -2 yields a zero in
the denominator of ( f − g ) . So, the domain
is {x | x ≠ -2, x ∈ �}.
Given f(x) = x2 - 3 and g(x) = 1 −
x , find each function and its domain.
a. (f - g)(x)
(f - g)x = f(x) - g(x)= x2 - 3 - 1 −
x
The domain of f is (-∞, ∞) and the domain of g is (−∞, 0) ∪ (0, ∞), so the domain of (f - g) is (−∞, 0) ∪ (0, ∞).
b. (f � g)(x)
(f � g)x = f(x) � g(x)
= (x2 - 3) 1 −
x
= x - 3 −
x
The domain of f is (-∞, ∞) and the domain of g is (−∞, 0) ∪ (0, ∞), so the domain of (f - g) is (−∞, 0) ∪ (0, ∞).
Exercises
Find (f + g)(x), (f - g)(x), (f � g)(x), and ( f −
g ) (x) for each f(x) and g(x).
State the domain of each new function.
1. f(x) = x2 - 1, g(x) = 2 −
x 2. f(x) = x2
+ 4x − 7, g(x) = √
x
Study Guide and InterventionFunction Operations and Composition of Functions
Compositions of Functions In a function composition, the result of one function is used to evaluate a second function.
Given functions f and g, the composite function f ◦ g can be described by the equation [f ◦ g](x) = f[g(x)]. The domain of f ◦ g includes all x-values in the domain of g for which g(x) is in the domain of f.
Given f(x) = 3x2 + 2x - 1 and g(x) = 4x + 2, find [f ◦ g](x) and [g ◦ f](x).
[f ◦ g](x) = f[g(x)] Defi nition of composite functions
= f(4x + 2) Replace g(x) with 4x + 2.
= 3(4x + 2)2 + 2(4x + 2) - 1 Substitute 4x + 2 for x in f(x).
= 3(16x2 + 16x + 4) + 8x + 4 - 1 Simplify.
= 48x2 + 56x + 15
[g ◦ f](x) = g(f(x)) Defi nition of composite functions
= g(3x2 + 2x - 1) Replace f(x) with 3x2 + 2x - 1.
= 4(3x2 + 2x - 1) + 2 Substitute 3x2 + 2x - 1 for x in g(x).
= 12x2 + 8x - 2 Simplify.
Exercises
For each pair of functions, find [f ◦ g](x), [g ◦ f](x), and [f ◦ g](4).
Find two functions f and g such that h(x) = [f ◦ g](x). Neither function may be the identity function f(x) = x.
9. h(x) = √ ��� 2x - 6 -1 10. h(x) = 1 −
3x +3
11. RESTAURANT A group of three restaurant patrons order the same meal and drink and leave an 18% tip. Determine functions that represent the cost of all of the meals before tip, the actual tip, and the composition of the two functions that gives the cost for all of the meals including tip.
PracticeFunction Operations and Composition of Functions
1. MARCHING BAND Band members form a circle of radius r when the music starts. They march outward as they play. The function f(t) = 2.5t gives the radius ofthe circle in feet after t seconds.Using g(r) = πr2 for the area of the circle, write a composite function that gives the area of the circle after t seconds.Then find the area, to the nearest tenth, after 4 seconds.
2. CANDLES A hobbyist makes and sells candles at a local market. The function c(h) = 4h gives the number of candles she has made after h hours. The function f(c) = 12 + 0.25c gives the cost of making c candles.
a. Write the composite function that gives the cost of candle making after h hours.
b. A sale reduces the cost of making c candles by 10%. Write the sale function s(x) and the composite function that gives the cost of candle making after h hours if materials are purchased during the sale.
3. SCIENCE The function t(x) = √ � 2x
−
28 + 6.25
gives the temperature in degrees Celsius of the liquid in a beaker after x seconds. Decompose the function into two separate functions, s(x) and r(x), so that s(r(x)) = t(x).
4. TRAVEL Two travelers are budgeting money for the same trip. The first traveler’s budget (in dollars) can be represented by f(x) = 45x + 350. The second traveler’s budget (in dollars) can be represented by g(x) = 60x + 475, x is the number of nights.
a. Find (f + g)(x) and the relevant domain.
b. What does the composite function in part a represent?
c. Find (f + g)(7) and explain what the value represents.
d. Repeat parts a–c for (g - f)(x).
5. POPULATION The function p(x) = 2x2
- 12x + 18 predicts the population of elk in a forest for the years 2010 through 2015 where x is the number of years since 2000. Decompose the function into two separate functions, a(x) and b(x), so that [a ◦ b](x) = p(x) and a(x) is a quadratic function and b(x) is a linear function.
Word Problem PracticeFunction Operations and Composition of Functions
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