Math 2B/3A

Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434

Name: Date:

 

  1  

Chapter 4 Quiz 2 Review Students will be able to…

• Determine whether a relation is a function based on a description, table, or graph. • Find the domain and range of a function from a graph. • Find the domain and range of linear, quadratic, exponential, rational, and radical (square

root) functions from their equations. • Evaluate or find a formula for a composition of functions. • Determine whether a function’s inverse will be a function by determining whether the

original function is one-to-one. • Find the inverse of a function, given its table, graph, or function formula.

Note: topics from the previous quiz are fair game. Expect at least one question on the quiz to

be review and use the review sheet from the last quiz to review those topics. Practice Problems: Complete these problems on a separate sheet of paper. You may need to

turn them in! 1. Determine whether each relation given below is a function, and explain why or why not. a. The input is a Math 2B/3A student, and the output is their math teacher. b. The input is a Math 2B/3A teacher, and the output is their student. c. The input is a day of the week, and the output is how many free blocks you have that day. d. The input is a student athlete at LHS and the output is a sport they play. e.

x f(x) 0 5 1 -6 1 -1 2 7 3 9

f.

x g(x) 1 -9 2 16 3 72 4 -11 5 0

g.

Student Favorite Subject

Tim math Sophie math Shant math Anna math

Marnier math

h.

i.

Math 2B/3A

Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434

Name: Date:

 

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2. State the domain and range of each function below.

a.

b.

c.

d. 𝑓 𝑥 = 2, 4 −3, 5 −7,4 (5, 0)

e. x f(x) −1 1 0 3 2 5 3 7 5 9

3. State the domain of each function below. a. f (x) = x2

b. g(x) = 3⋅2x c. y =18x − 47

d. h(x) = 2x −14

e. f (x) = 54x −16

f. f (x) = 3x +8

4. Let f (x) = −4x + 7 and g(x) = x −12

. Find each of the following.

a. f (2) b. g(3) c. f (g(−3)) d. g( f (4)) e. f (g(a)) f. g( f (a)) g. f ( f −1(x)) h. g(g−1(x))

6

4

2

–2

–4

–6

–5 5

Math 2B/3A

Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434

Name: Date:

 

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5. Let R(x) = x − 2 and Q(x) = x2 + 4x +1 . a. Find a formula for R(Q(x)) . b. Find a formula for Q(R(x)) .

6. Let h(x) = 4x +1 and j(x) = x −14

.

a. Find a formula for h( j(x)) . b. Find a formula for j(h(x)) .     c. What do the above results tell you about the relationship between h(x) and j(x)? 7. Determine whether each function is one-to-one. If it is, find the inverse function. If not,

explain why not.

a.

x f(x) 0 -6 1 -5 2 -2 3 3 4 10

b.

x g(x) 1 3 2 1 3 -2 4 1 5 10

c.

Student Favorite Subject

Tim math Sophie math Shant math Anna math

Marnier math

d.

e.

6

4

2

–2

–4

–6

–5 5

Math 2B/3A

Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434

Name: Date:

 

  4  

f.

g.

8. Find the inverse of each function. Show your work. a. f (x) = 2x − 7

b. h(x) = 3− x2

c. h(x) = 3x + 49

d. f (x) = −5x +1

e. g(x) = x + 6−4

9. George is shopping for a new bed. There is a $50 delivery fee for the bed. George also must pay 6.5% sales tax.

a. Write an expression 𝑡(𝑥) for the cost of the bed with sales tax (but without the delivery charge) for a bed costing 𝑥 dollars.

b. Write an expression 𝑑 𝑥 for the cost of the bed with a delivery charge (but no tax.) c. Find 𝑡 𝑑 𝑥 and 𝑑 𝑡 𝑥 d. Which of these compositions describes the total cost of the bed if tax is charged on

delivery?

Math 2B/3A

Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434

Name: Date:

 

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Answers 1. a. Function – Each student has exactly one math teacher, so each input has exactly one

output. b. Not a function – Each teacher has lots of students, so each input doesn’t have exactly one

output. c. Function – On a given day you have a set number of free blocks, so each input (day of the

week) matches with exactly one output (number of free blocks). d. Not a function – A student could play multiple sports, so an input could have multiple

outputs. e. Not a function – The input 1 has two different outputs. f. Function – Each input has exactly one output. g. Function – Each student (input) has exactly one favorite subject (output). A function can

have multiple inputs having the same output, so it’s okay that all the students have math as a favorite subject. J

h. Function – Each x value has a single y value. There aren’t any points on the graph that have the same x value but different y values.

i. Not a function – The x values 1 and -1 each have two different y values. 2. a. domain: x ∈ R, − 4 ≤ x < 5 , range: y ∈ R, − 2 ≤ y < 8 b. domain: x ∈ R, 0 < x ≤ 7 , range: y ∈ R, − 6 ≤ y ≤10 c. domain: x ∈ R, -7 < x ≤ 7 , range: y ∈ R, −3≤ y ≤ 6 3. a. domain: x ∈ R b. domain: c. domain: d. domain: x ∈ R, x ≥ 7 e. domain: x ∈ R, x ≠ 4 f. domain: 4. a. f (2) = −1 b. g(3) =1 c. f (g(−3)) =15 d. g( f (4)) = −5

e. f (g(a)) = −4 a−12

"

#$

%

&'+ 7 f. g( f (a)) = (−4a+ 7)−1

2

f (g(a)) = −4a+ 42

+ 7

f (g(a)) = −2a+ 2+ 7f (g(a)) = −2a+ 9

   

g( f (a)) = −4a+ 7−12

g( f (a)) = −4a+ 62

g( f (a)) = −2a+3

 

g. f ( f −1(x)) = x h. g(g−1(x)) = x

x ∈ Rx ∈ R

x ∈ R

Math 2B/3A

Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434

Name: Date:

 

  6  

5. a. R(Q(x)) = R(x2 + 4x +1)

R(Q(x)) = x 2+4x +1− 2R(Q(x)) = x 2+4x −1

b. Q(R(x)) =Q(x − 2)

Q(R(x)) = (x − 2)2 + 4(x − 2)+1Q(R(x)) = x2 − 4x + 4+ 4x −8+1Q(R(x)) = x2 −3

6. a.

€

h( j(x)) = hx −14

#

$ %

&

' ( b.

€

j(h(x)) = j(4x +1)

€

h( j(x)) = 4 x −14

#

$ %

&

' ( +1

h( j(x)) = x −1+1h( j(x)) = x

€

j(h(x)) =4x +1−1

4

j(h(x)) =4x4

j(h(x)) = x

7. a. f(x) is one-to-one, Inverse: b. g(x) is not one-to-one, because both the inputs 2 and 4 have an output of 1. c. This function is not one-to-one because all the inputs (students) have the same output

(favorite subject – math of course!). Therefore, the inverse will not be a function. d. This function is not one-to-one because there are lots of places where you have two points

with the same y value but different x values (for example, the points (1, 1) and (-6, 1) are both on the graph). Therefore, the inverse will not be a function.

e. One-to-one. f. One-to-one g. One-to-one

x f-1(x) -6 0 -5 1 -2 2 3 3 10 4

Math 2B/3A

Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434

Name: Date:

 

  7  

8. a. f (x) = 2x − 7 f: 1. Multiply by 2. 2. Subtract 7. f -1: 1. Add 7. 2. Divide by 2.

€

f −1(x) =x + 72

€

f ( f −1(x)) = x

2⋅ f −1(x) − 7 = x

2⋅ f −1(x) = x + 7

f −1(x) =x + 72

b. h(x) = 3− x2

h: 1. Multiply by -1. 2. Add 3. 3. Divide by 2 h -1: 1. Multiply by 2. 2. Subtract 3. 3. Divide by -1.

€

h−1(x) =2x − 3−1

h−1(x) = −(2x − 3)h−1(x) = −2x + 3

€

h(h−1(x)) = x

3 − h−1(x)2

= x

3 − h−1(x) = 2x−h−1(x) = 2x − 3h−1(x) = −2x + 3

c. h(x) = 3x + 49

h: 1. Multiply by 3. 2. Add 4. 3. Divide by 9. h -1: 1. Multiply by 9. 2. Subtract 4. 3. Divide by 3.

€

h−1(x) =9x − 43

€

h(h−1(x)) = x

3⋅ h−1(x) + 49

= x

3⋅ h−1(x) + 4 = 9x3⋅ h−1(x) = 9x − 4

h−1(x) =9x − 43

Math 2B/3A

Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434

Name: Date:

 

  8  

d. f (x) = −5x +1 f: 1. Multiply by -5. 2. Add 1. f -1: 1. Subtract 1. 2. Divide by -5.

€

f −1(x) =x −1−5

€

f ( f −1(x)) = x

−5⋅ f −1(x) +1 = x

−5⋅ f −1(x) = x −1

f −1(x) =x −1−5

e. g(x) = x + 6−4

g: 1. Add 6. 2. Divide by -4. g -1: 1. Multiply by -4. 2. Subtract 6.

€

g−1(x) = −4x − 6

€

g(g−1(x)) = x

g−1(x) + 6−4

= x

g−1(x) + 6 = −4xg−1(x) = −4x − 6

 9.    

a. 𝑡 𝑥 = 1.065𝑥  b. 𝑑 𝑥 = 𝑥 + 50  c. 𝑡 𝑑 𝑥 = 1.065𝑥 + 53.25  

𝑑 𝑡 𝑥 = 1.065𝑥 + 50  d. 𝑡 𝑑 𝑥  represents  paying  tax  on  the  delivery  charge,  𝑑 𝑡 𝑥  represents  no  tax  on  

the  delivery  charge.    10.    

a.