Deck 9: Exponential and Logarithmic Functions

Question 1

Let $f ( x ) = 3 x - 4$ and $g ( x ) = x ^ { 2 } - 1$ . Find: $( f \circ g ) ( x )$

A) $3 x ^ { 2 } - 5$
B) $3 x ^ { 2 } - 7$
C) $9 x ^ { 2 } - 24 x + 15$
D) $3 x ^ { 2 } - 3$

Answer 1

$3 x ^ { 2 } - 7$

Question 2

Let and . Find the function and its domain.

Answer 2

The domain is .

Question 3

If and __________, then

Answer 3

To find the function $g(x)$ such that $(g \circ f)(x) = 3x - 1$ , we need to understand that the composition of functions $(g \circ f)(x)$ means applying the function $f$ to $x$ first, and then applying the function $g$ to the result of $f(x)$ .

Given $f(x) = \sqrt{x}$ , we want to find $g(x)$ such that when we apply $g$ to $f(x)$ , we get $3x - 1$ .

Let's denote $y = f(x) = \sqrt{x}$ . Then, we want to find $g(y)$ such that $g(y) = 3x - 1$ .

Since $y = \sqrt{x}$ , we can square both sides to find $x$ :

$y^2 = (\sqrt{x})^2$
$y^2 = x$

Now, we substitute $y^2$ for $x$ in the equation $g(y) = 3x - 1$ :

$g(y) = 3y^2 - 1$

Therefore, the function $g(x)$ that satisfies the condition $(g \circ f)(x) = 3x - 1$ is:

$g(x) = 3x^2 - 1$

Question 4

If __________, then

Answer 4

To find the function $g(x)$ such that $(f \circ g)(x) = x^3 + 9x^2 + 27x + 27$ , we need to determine what function $g$ when composed with $f(x) = x^3$ results in the given polynomial.

Given that $f(x) = x^3$ , the composition $(f \circ g)(x)$ means we are taking the output of $g(x)$ and plugging it into $f$ . In other words, we are cubing whatever $g(x)$ is. So, we are looking for a function $g(x)$ such that when we cube it, we get $x^3 + 9x^2 + 27x + 27$ .

Let's assume $g(x) = ax^2 + bx + c$ . Then we have:

$$f(g(x)) = (ax^2 + bx + c)^3$$

We need to expand this and match it to the given polynomial $x^3 + 9x^2 + 27x + 27$ . However, we can notice that the given polynomial can be factored as:

$$x^3 + 9x^2 + 27x + 27 = (x + 3)^3$$

This suggests that $g(x)$ should be $x + 3$ , because when we plug $x + 3$ into $f(x) = x^3$ , we get:

$$f(g(x)) = f(x + 3) = (x + 3)^3 = x^3 + 9x^2 + 27x + 27$$

Therefore, the function $g(x)$ that satisfies the given condition is:

$$g(x) = x + 3$$

Question 5

Let $f ( x ) = 7 x - 6 \text { and } g ( x ) = - 2 x + 2$ . Find: $f + g$

A) $- 14 x ^ { 2 } + 26 x - 12$
B) $5 x + 4$
C) $5 x - 4$
D) $5 x - 8$

Question 6

The __________ of and , denoted as , is defined by __________.

Question 7

Let $f ( x ) = 3 x - 4$ and $g ( x ) = x ^ { 2 } - 1$ . Find: $( g \circ f ) ( - 1 )$

A) -4
B) 0
C) $- 3 x ^ { 2 } + 24 x - 15$
D) 48

Question 8

Let and . Find the composition.

Question 9

Let $f ( x ) = 5 x - 5 \text { and } g ( x ) = - 2 x + 9$ . Find: $( f - g ) ( 3 )$ .

A) 13
B) 7
C) 20
D) 30

Question 10

Let $f ( x ) = 5 x - 4$ and $g ( x ) = x ^ { 2 } - 6$ . Find: f g

A) $- x ^ { 2 } + 5 x + 2$
B) $- x ^ { 2 } + 5 x - 10$
C) $x ^ { 2 } - 5 x - 2$
D) $- x ^ { 2 } + 5 x - 2$

Question 11

The __________ of the function is the set of real numbers x that are in the domain of both f and g .

Question 12

Let $f ( x ) = 5 x - 9$ and $g ( x ) = 4 x - 2$ . Find the domain of $\frac { f } { g }$ .

A) $( - \infty , 0 ) \cup ( 0 , \infty )$
B) $( - \infty , \infty )$
C) $\left( - \infty , \frac { 9 } { 5 } \right) \cup \left( \frac { 9 } { 5 } , \infty \right)$
D) $\left( - \infty , \frac { 1 } { 2 } \right) \cup \left( \frac { 1 } { 2 } , \infty \right)$

Question 13

Let and . Find the function and its domain.

Question 14

Let and . Find the composition.

Question 15

Let $f ( x ) = 3 x - 5$ and $g ( x ) = x ^ { 2 } - 3$ . Find: f . g

A) $3 x ^ { 2 } - 15$
B) $3 x ^ { 3 } + 15$
C) $3 x ^ { 3 } - 5 x ^ { 2 } - 9 x + 15$
D) $3 x ^ { 3 } - 5 x ^ { 2 } - 9 x - 15$

Question 16

Let $f ( x ) = 4 x - 7 \text { and } g ( x ) = - 2 x + 9$ Find: $( f \cdot g ) ( 1 )$

A) 21
B) -21
C) -16
D) 0

Question 17

The __________ of and , denoted as , is defined by __________.

Question 18

Let and . Find the composition.

Question 19

Let and . Find the composition.

Question 20

The __________ function is defined by __________.

Question 21

Let and . Find the composition.

Question 22

The __________ function is defined by __________.

Question 23

The __________ of and , denoted as , is defined by __________ and the __________ of and , denoted as , is defined by __________.

Question 24

The __________ of and , denoted as , is defined by __________ and the __________ of and , denoted as , is defined by __________.

Question 25

Let and . Find the composition.

Question 26

Let and . Find the composition.

Question 27

Determine whether the function $f ( x ) = | x + 5 |$ is one-to-one.

A) one-to-one
B) not one-to-one

Question 28

Let and . Find the function .

Question 29

The graphs of a function and its inverse are _________ to the line

Question 30

The __________ line test can be used to decide whether the graph of a function represents a one-to-one function.

Question 31

Let and . Find the composition.

Question 32

If the function $f ( x ) = 9 x - 4$ is one-to-one, find its inverse.

A) $f ^ { - 1 } ( x ) = \frac { x - 4 } { 9 }$
B) $f ^ { - 1 } ( x ) = \frac { 1 } { 9 x - 4 }$
C) $f ^ { - 1 } ( x ) = \frac { x + 4 } { 9 }$
D) not one-to-one

Question 33

A function is called a __________ function if different inputs determine different outputs.

Question 34

Let and . Find the composition.

Question 35

If the point (- 9, 2) is on the graph of the one-to-one function f , then what point is on the graph of f ^-1?

A)
B)
C)
D)

Question 36

The __________ line test can be used to decide whether the graph of a function represents a one-to-one function.

Question 37

A function is called a __________ function if different inputs determine different outputs.

Question 38

Let and . Find the domain of the function .

Question 39

Determine the graph of the function is one-to-one.

A) not one-to one
B) one-to one

Question 40

Let and . Find the function .

Question 41

Let and . Find the composition.

Question 42

Let and . Find the domain of the function .

Question 43

If the function $f ( x ) = \frac { x + 9 } { 5 }$ is one-to-one, find its inverse.

A) $f ^ { - 1 } ( x ) = \frac { 5 } { x + 9 }$
B) $f ^ { - 1 } ( x ) = 5 x - 9$
C) $f ^ { - 1 } ( x ) = 5 x + 9$
D) not one-to-one

Question 44

Let and . Find the composition.

Question 45

Select the graph of the function that is one-to-one.

A)
B)
C)
D)

Question 46

Let and . Find the domain of the function .

Question 47

The functions are __________.

Question 48

Let and . Find the function .

Question 49

Which of the following statements is true for a one-to-one function.

A) A vertical line intersects the graph in more than one point.
B) A horizontal line intersects the graph in more than one point.
C) A horizontal line intersects the graph in at most one point.
D) none of these

Question 50

Let and . Find the composition.

Question 51

The graph represents a function. Use the horizontal line test to decide whether the function is one-to-one. Answer yes or no .

Question 52

Let and . Find the composition.

Question 53

Determine whether the function is one-to-one. Answer yes or no .

Question 54

Let and . Find the composition.

Question 55

Find the inverse of the function and express it using notation.

Question 56

Let and . Find the composition.

Question 57

The graph represents a function. Use the horizontal line test to decide whether the function is one-to-one. Answer yes or no .

Question 58

Let and . Find the composition.

Question 59

If the point is on the graph of the one-to-one function , then the point __________ is on the graph of .

Question 60

If is a one-to-one function, the domain of is the __________ of , and the range of is the __________ of .

Question 61

Determine whether the function is one-to-one.

Question 62

Determine whether the function is one-to-one. Answer yes or no .

Question 63

If is a one-to-one function, the domain of is the __________ of .

Question 64

Determine whether the function is one-to-one. Answer yes or no .

Question 65

Determine whether the function is one-to-one.

Question 66

Determine whether the function is one-to-one. Answer yes or no .

Question 67

Which of the following statements is true for $y = 9 ^ { x }$ ?

A) The graph of $y = 9 ^ { x }$ is an increasing function.
B) The y -axis is a horizontal asymptote of the graph.
C) The range is all real numbers.
D) (1, 0) is a point of the graph of $y = 9 ^ { x }$ .

Question 68

Graph the inverse of the following one-to-one function.

Question 69

Exponential functions have a constant base and a variable _________.

Question 70

Compare the graph of $f ( x ) = 6 ^ { x - 5 }$ to the graph of $y = 6 ^ { x }$ .

A) The graph of $f ( x ) = 6 ^ { x - 5 }$ is shifted 5 units to the left of the graph of $y = 6 ^ { x }$ .
B) The graph of $f ( x ) = 6 ^ { x - 5 }$ is shifted 5 units to the right of the graph of $y = 6 ^ { x }$ .
C) The graph of $f ( x ) = 6 ^ { x - 5 }$ is shifted up 5 units from the graph of $y = 6 ^ { x }$ .
D) The graph of $f ( x ) = 6 ^ { x - 5 }$ is shifted down 5 units from the graph of $y = 6 ^ { x }$ .

Question 71

Two exponential functions of the form $f ( x ) = b ^ { x }$ are graphed below. Which function has the larger base b , the one graphed in red or the one graphed in blue?

A) red
B) blue

Question 72

Determine whether the function is one-to-one.

Question 73

An initial deposit of $20,000 earns 3% interest, compounded monthly. How much will be in the account after 15 years? Round your answer to the nearest cent.

A) $4,090,067.19
B) $31,348.63
C) $31,159.35
D) $20,763.27

Question 74

An initial deposit of $20,000 earns 8% interest, compounded quarterly. How much will be in the account after 10 years?
Please give the answer to the nearest cent. $__________

Question 75

Determine whether the function is one-to-one.

Question 76

Find the inverse of the function:

Question 77

If the point is on the graph of the one-to-one function , then what point is on the graph of ?

Question 78

Use composition to show that the pair of functions are inverses.

Question 79

Find the inverse of the function and express it using notation.

Question 80

Find the inverse of the function: