Problem set 11 & 12: Operations on Functions and Inverse Functions
1. Consider the following functions:
f ( x) = 9 − 6 x , g ( x) = x + 7
Find (f + g)(x), (f − g)(x), (fg)(x), and (f/g)(x), and their domains.
2. Consider the functions: f(x) = 4x - 6, g(x) = 2x2 - x + 7
(a) Find (f
g)(x)
(b) Find (g
f)(x)
(c) Find (f(g(-2))
(d) Find (g(f(3))
3. Consider the following functions: f(x) = |6x|,
(a) Find (f
g)(x).
(b) Find (g
f)(x).
(d) Find g(f(3)).
g(x) = −1
(c) Find f(g(−2)).
4. Several values of two functions T and S are listed in the tables.
t
5
3
7
8
4
T(t)
7
8
3
5
9
x
5
3
7
8
4
S(x)
Find the
(a) (S
(b) (T
(c) (S
(d) (T
3
5
8
7
9
expressions, if possible. (If it is not possible, enter NONE.)
T)(3)=
T)(3)=
S)(3)=
S)(4)=
5. Consider the functions:
(a) Find (f
f ( x) =
x
5
, g ( x) =
x−4
x
g)(x) and the domain of f
6. Consider the following functions:
(a) Find (f
g
(b) Find (g
f)(x) and the domain of g
f.
f)(x) and the domain of g
f
f ( x) = x 2 + 33, g ( x) = 3 x − 33
g)(x) and the domain of f
g
(b) Find (g
7. Find functions f, g such that f(g(x)) = y, where
y = 3 + x2 + 6
(a) g ( x) = 3 + x + 6 , f ( x) = x 2 + 6
(b) g ( x) = 3 + x , f ( x) = x 2 + 6
(c ) g ( x ) = 3 + x 2 , f ( x ) = x + 6
(d ) g ( x) = x 2 + 3, f ( x) = 6 + x
(e) g ( x) = x 2 + 6, f ( x) = 3 + x + 6
( f ) g ( x) = 6 + x , f ( x) = x 2 + 3
( g ) g ( x) = x 2 + 6, f ( x) = 3 + x
(h) g ( x) = x + 6, f ( x) = 3 + x 2
8. Solve the equation (f
g)(x) = 0, where f(x) = x2 − 2,
g(x) = x + 4
The bottom graph above is of the function f(x), and the top graph is g(x). Use the graphs to compute
the following function values. The spacing between each gridline is one unit.
(a) f(g(3))
(b) f(g(0))
9. The functions f, g, and h are defined below:
f(x) = 9x + 8, g(x) = x2 + 2x + 6, h(x) = 1 - 8x2
Classify each of the following compositions of functions as linear, quadratic, or neither.
f(g(x))
f(f(x))
h(g(x))
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Problem set 11 & 12: Operations on Functions and Inverse Functions
10. It has been estimated that 1000 curies of a radioactive substance introduced at a point on the
surface of the open sea would spread over an area of 40,000 km2 in 40 days. This means that the
function g(t) = 1000t represents the contaminated area as a function of time, the number of days
since contamination. From geometry, you can create a function r = f(A) that expresses the radius of a
circle as a function of area.
(a) Express the radius as a function of A (upper case!): f(A) =
(b) what does the composite function
model?
o The area of the contamination as a function of time
o The area of the contamination as a function of radius
o The radius of the contamination as a function of time
o The radius of the contamination as a function of area
11. Determine whether the following functions are one-to-one:
f ( x) = 3 x , g ( x) =| 2 x + 1 |-3
12. Find the inverse function of f(x) = 7x3 − 5
13. Find the inverse function of
f ( x) =
3x + 2
2x − 9
14. Find the inverse function of f: f(x) = 6 − 5x2,
x≤0
15. Use the theorem on inverse functions to determine whether f and g are inverse functions of each
other:
f ( x) = − x 2 + 8, x ≥ 0 & g ( x) = 8 − x , x ≤ 8
Sketch the graphs of f and g on the same coordinate plane.
16. Determine the domain and range of f −1 for the given function without actually
finding f−1. (Hint: First find the domain and range of f):
f ( x) =
4
x+7
17. Use the table for f(x) compute each expression. Enter NONE if there isn't any value to report.
(a) f -1(3) =
x 2
3 1 4
5
(b) f -1(4) =
f(x) -1 6 4 2
3
(c) f -1(2) =
18. The point (a,b) is on the graph of the one-to-one function y = f(x). For each of the following
functions, enter the ordered pair that corresponds to the transformation of (a,b). For example, the
graph of y = f(x) + 1 is obtained by translating the graph of y = f(x) up one unit so the corresponding
point on the new graph is (a,b + 1).
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Problem set 11 & 12: Operations on Functions and Inverse Functions
19. Several values of two functions f and g are listed in the tables. Both f and g are one-to-one
functions.
T
3
2
9
1
4
f(t)
1
9
2
3
5
X
3
2
9
1
4
g(x)
2
3
5
9
1
Find the expressions, if possible. (If it is not possible, enter NONE.)
(a) (f
g)(1)=
(b) (g
f)(2)=
(c) (f
f)(3) =
(d) (g
f -1)(3) =
(e) (g
g)(9) =
20. Let h(x) = 4 − x. Use h, the table, and the graph to evaluate the expression.
X
f(x)
(a) (g−1
(b) (g−1
(c) (h−1
2
3
4
5
6
−1
0
1
2
3
f −1)(2)
h)(3)
f
g−1)(3)
21. The graph of a one-to-one function f is shown.
(a) Use the reflection property to sketch the graph of f −1.
(b) Find the domain D and range R of the function f.
(c) Find the domain D1 and range R1 of the inverse function f −1.
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Problem set 11 & 12: Operations on Functions and Inverse Functions
22. Ventilation is an effective way to improve indoor air quality. In nonsmoking restaurants, air
circulation requirements (in ft3/min) are given by the function V(x) = 35x, where x is the number of
people in the dining area.
(a) Determine the ventilation requirements for 27 people = ft3/min
(b) Find V −1(y) (we don't change variables because they hold different meaning). Explain the
significance of V −1.
o V −1 computes the ventilation requirements when there are no people in the restaurant.
o V −1 computes the amount of space, in ft3, it would take to fulfill the ventilation requirements
for x people.
o V −1 computes the reduction in ventilation requirements, in ft3/min, when x people leave the
restaurant.
o V −1 computes the minimum number of people that should be in the restaurant at one time. V
−1
computes the maximum number of people that should be in the restaurant at one time.
(c) Use V −1 to determine the maximum number of people that should be in a restaurant having a
ventilation capability of 2400 ft3/min. (Round your answer down to the nearest whole number.)
23.
(a) Temperature scales in degrees Celsius and Fahrenheit are related linearly. Useful temperature data
is that 100 degrees Celsius corresponds to 212 degrees Fahrenheit (water boils) and 0 degrees Celsius
corresponds to 32 degrees Fahrenheit (water freezes). Find a linear function g that inputs a
temperature x in degrees Celsius and outputs the temperature converted to degrees
Fahrenheit, g(x). g(x) =
(b) Interpret the meaning of the function g-1
o The temperature input in degrees Fahrenheit is converted to degrees Celsius
o The temperature input in degrees Fahrenheit is converted to degrees Celsius
o The temperature input in degrees Celsius is converted to degrees Fahrenheit.
o The temperature input is inverted.
(c) Wind chill temperature is a measure of how cold it actually feels on human skin, when outdoor
temperature combines with the cooling effect of wind. When the wind speed is constant, we can model
a linear function f which inputs the current temperature in degrees Fahrenheit and outputs the wind
chill temperature ("how cold you feel"). Given that we can convert a temperature reading in degrees
Celsius to Fahrenheit via your function g, what does the composition f g represent?
o The wind chill input is converted to actual temperature in degrees Celsius.
o The wind chill input in degrees Celsius is converted to the wind chill in degrees Fahrenheit
o The temperature input in degrees Fahrenheit is converted to wind chill in degrees Celsius.
o The temperature input in degrees Celsius is converted to the wind chill in degrees Fahrenheit.
o The wind chill input in degrees Fahrenheit is converted to the wind chill in degrees Celsius.
(d) What type of function is f g? Linear, Quadratic, Neither
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