Review for Precalculus Test #2: Functions
Here are some review problems for this week’s test. You should do these problems and study your notes, homework and
quizzes from this unit.
1. Is y a function of x? Explain your answer.
a. 𝑦 = 𝑥 2 + 2
b. 𝑦 = |𝑥 − 1|
c. 𝑥 = 𝑦 2 + 2
d. y = f(x) (see graph)
2. Find the domain and zeros of each function:
a. 𝑦 = √9 − 𝑥
b. 𝑦 = √𝑥 2 − 9
𝑥+2
c. 𝑦 = 𝑥−2
d. 𝑦 =
𝑥 2 −5𝑥+6
4𝑥 2 −1
3. Find the domain, range and zeros of y = f(x). (see graph)
4. Consider the functions 𝑓(𝑥) = 𝑥 2 − 1 and (𝑥) = 2𝑥 2 . Write and simplify an expression for the following.
a. (𝑓 + 𝑔)(𝑥)
e. (𝑓𝑜𝑔)(𝑥)
b. (𝑓 − 𝑔)(𝑥)
f. (𝑔𝑜𝑓)(𝑥)
c. (𝑓𝑔)(𝑥)
𝑓
𝑔
d. ( ) (𝑥) What is the domain of this function?
5. Use the tables below to find the following function values.
-1
0
1
2
3
𝒙
𝒇(𝒙)
0
1
2
3
4
𝒙
𝒈(𝒙)
1
-1
2
1
𝑓
a. (𝑓 + 𝑔)(3)
b. (𝑓 − 𝑔)(1)
c. (𝑓𝑔)(2)
3
3
4
5
g. (𝑔𝑜𝑔)(2)
h. 𝑓 −1 (3)
i. 𝑔−1 (1)
d. (𝑔) (2)
e. (𝑓𝑜𝑔)(1)
f. (𝑔𝑜𝑓)(3)
6. Use the graphs to find the following function values.
a. (𝑔 − 𝑓)(4)
b. (𝑔𝑓)(0)
c. (𝑓𝑜𝑔)(2)
d. (𝑔𝑜𝑓)(4)
e. 𝑓 −1 (−3)
1
1
7. Let 𝑓(𝑥) = 𝑥 2 and 𝑔(𝑥) = 𝑥−1 .
a.
b.
c.
d.
Find 𝑓𝑜𝑔(𝑥) and 𝑔𝑜𝑓(𝑥).
What is the domain of 𝑓𝑜𝑔?
Does 𝑓 have an inverse that is a function? How do you know? If an inverse exists, find 𝑓 −1 .
Does 𝑔 have an inverse that is a function? How do you know? If an inverse exists, find 𝑔−1 .
8. a. Draw the graph of a function whose inverse is also a function.
b. Draw the graph of a function whose inverse is not a function.
c. Draw a graph of a relation that is not a function.
9. Suppose h is a function whose inverse, ℎ−1, is also a function. What is ℎ𝑜ℎ−1 (12)?
10. For letters a-c, find two functions 𝑓 and 𝑔 such that ℎ(𝑥) = (𝑓𝑜𝑔)(𝑥)
a. ℎ(𝑥) = √9 − 𝑥
b. ℎ(𝑥) =
4
(5𝑥+2)2
11. The total cost of renting a car is 1000 pesos plus 15 pesos per mile and can be calculated by the function
C(x) = 1000 + 15x.
a. What is 𝐶(50), and what does it represent in the context of this problem?
b. Find an equation for 𝐶 −1 (𝑥).
c. What is 𝐶 −1 (1975) and what does it represent in the context of this problem?
12. Find the inverse of 𝑓(𝑥) =
𝑋+1
2
and graph both functions on the same axes.
13. If 𝑓(𝑥) = 𝑥 2 − 3 with domain 𝑥 ≤ 0, find the inverse of f(x) and state its domain and range.
14. If𝑔(𝑥) = (𝑥 − 4)2 with domain 𝑥 ≥ 4, find the inverse of g(x) and state its domain and range.
15. The graph shown is y = f(x). Graph y = -f(x+2)
16. If f(x) = x2 and g(x) = f(x-1) +2,
a. Write and simplify an equation for g(x) that is not in terms of f.
b. Describe the transformation of the graph of f into the graph of g.