Question 1

Find a formula for the inverse of the function described below.
-$32 ^ { \circ }$ Fahrenheit $= 0 ^ { \circ }$ Celsius. A function that converts temperatures in Celsius to those in Fahrenheit is $f ( x ) = \frac { 9 } { 5 } x + 32$

Accepted Answer

To find the inverse function, $f^{-1}(x)$, you solve the equation $y = \frac{9}{5}x + 32$ for $x$, which gives $x = \frac{5}{9}(y - 32)$. Replace $y$ with $x$ to get the inverse function, $f^{-1}(x) = \frac{5}{9}(x - 32)$.

Question 2

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of
the inverse.
-$$f ( x ) = - 7 x - 4$$

Accepted Answer

The function $f(x) = -7x - 4$ is a linear function, which means its domain and range are all real numbers. Since it is a one-to-one function, the domain and range of its inverse are also all real numbers.

Question 3

Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
-The profit P for selling x items is given by the equation P(x) = 2x - 500. Express the sales amount x as a function
of the profit P.

Accepted Answer

The answer of Write the word or phrase that best...

Question 4

Approximate the value using a calculator. Express answer rounded to three decimal places.
-$3 ^ { 3.3 }$

Accepted Answer

Using a calculator to compute $3^{3.3}$, the result is approximately 37.541 when rounded to three decimal places.

Question 5

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of
the inverse.
-$$f ( x ) = \frac { 3 } { x - 1 }$$

Accepted Answer

The function $f(x) = \frac{3}{x-1}$ is undefined at $x = 1$, so its domain excludes 1. The range excludes 0 because there's no value of $x$ that will make $f(x) = 0$. For the inverse function, the roles of domain and range are reversed.

Question 6

Find a formula for the inverse of the function described below.
-A size 44 dress in Country C is size 2 in Country D. A function that converts dress sizes in Country C to those in Country D is $$f ( x ) = \frac { x } { 2 } - 20$$

Accepted Answer

The answer of Find a formula for the inverse of...

Question 7

The function f is one-to-one. Find its inverse.
-$$f ( x ) = \sqrt [ 3 ] { x + 5 }$$

Accepted Answer

To find the inverse of $f(x) = \sqrt[3]{x + 5}$, you swap $x$ and $y$ and solve for $y$. So, $x = \sqrt[3]{y + 5}$ becomes $x^3 = y + 5$, which simplifies to $y = x^3 - 5$. Therefore, $f^{-1}(x) = x^3 - 5$.

Question 8

Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
-To remodel a bathroom, a contractor charges $\$ 25$ per hour plus material costs, which amount to $\$ 3,775$. Therefore, the total cost to remodel the bathroom is given by $f ( x ) = 25 x + 3,775$ where $x$ is the number of hou the contractor works. Find a formula for $f ^ { - 1 } ( x )$. What does $f ^ { - 1 } ( x )$ compute?

Accepted Answer

The inverse function $f^{-1}(x)$ is used to find the number of hours worked given the total cost $x$, and the correct formula is $f^{-1}(x) = \frac{x}{25} - 151$.

Question 9

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of
the inverse.
-$$f ( x ) = \frac { 4 } { 6 x + 5 }$$

Accepted Answer

The domain of $f(x)$ excludes the value that makes the denominator zero, which is $-\frac{5}{6}$. The range of $f(x)$ is all real numbers except 0, as the fraction cannot equal 0. The domain and range of the inverse function swap roles, hence the domain of $f^{-1}(x)$ excludes 0, and its range excludes $-\frac{5}{6}$.

Question 10

Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
-The function $f ( x ) = | x | - 5$ is not one-to-one.
(a) Find a suitable restriction on the domain of $\mathrm { f }$ so that the new function that results is one-to-one.
(b) Find the inverse of $f$.

Accepted Answer

The answer of Write the word or phrase that best...

Question 11

The function f is one-to-one. Find its inverse.
-$$f ( x ) = \frac { 3 x - 6 } { 8 x - 8 }$$

Accepted Answer

The answer of The function f is one-to-one. Find its...

Question 12

Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
-The weight W of a bird's brain (in ounces) is related to the volume V of the bird's skull (in cubic ounces)
through the function W(V) $$W ( V ) = 3.55 \sqrt [ 3 ] { V } + 1.23$$ (a) Express the skull volume V as a function of brain weight W.
(b) Predict the skull volume of a bird whose brain weighs 3 oz.

Accepted Answer

The answer of Write the word or phrase that best...

Question 13

Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
-The accompanying tables represent a function f that converts seconds to hours and a function g that converts hours to days. $\begin{array}{r|rrrrr}
\mathrm{x} & 86,400 & 172,800 & 259,200 & 345,600 & 432,000 \
\hline \mathrm{f}(\mathrm{x}) & 24 & 48 & 72 & 96 & 120
\end{array}$

$\begin{array}{r|rrrrr}
\mathrm{x} & 24 & 48 & 72 & 96 & 120 \
\hline \mathrm{g}(\mathrm{x}) & 1 & 2 & 3 & 4 & 5
\end{array}$

Express $\left( \mathrm { f } ^ { - 1 } \mathrm { gg } ^ { - 1 } \right) ( \mathrm { x } )$ symbolically.

Accepted Answer

The answer of Write the word or phrase that best...

Question 14

Find a formula for the inverse of the function described below.
-An organization determines that the cost per person of chartering a bus is given by the formula $$C ( x ) = \frac { 200 + 6 x } { x }$$ where x is the number of people in the group and C(x) is in dollars.

Accepted Answer

The answer of Find a formula for the inverse of...

Question 15

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of
the inverse.
-$$f ( x ) = \sqrt { 5 x + 4 }$$

Accepted Answer

The domain of $f(x) = \sqrt{5x + 4}$ is $x \geq -\frac{4}{5}$ because the expression under the square root must be non-negative. The range of $f(x)$ is $y \geq 0$ since square roots are always non-negative. For the inverse function, $f^{-1}(x)$, the domain and range swap roles, so the domain is $x \geq 0$ and the range is $y \geq -\frac{4}{5}$.

Question 16

Find a formula for the inverse of the function described below.
-A size 4 dress in Country C is size 40 in Country D. A function that converts dress sizes in Country C to those in Country D is f(x) = 2(x + 16).

Accepted Answer

To find the inverse, solve the equation $y = 2(x + 16)$ for $x$. This gives $x = \frac{y}{2} - 16$, so the inverse function is $f^{-1}(x) = \frac{x}{2} - 16$.

Question 17

Find a formula for the inverse of the function described below.
-A size 4 dress in Country $C$ is size 42 in Country D. A function that converts dress sizes in Country $C$ to those in Country $D$ is $f ( x ) = x + 38$.

Accepted Answer

To find the inverse function, $f^{-1}(x)$, we solve the equation $y = x + 38$ for $x$, which gives $x = y - 38$. Thus, the inverse function is $f^{-1}(x) = x - 38$.

Question 18

The function f is one-to-one. Find its inverse.
-$$f ( x ) = \sqrt { x - 1 }$$

Accepted Answer

The answer of The function f is one-to-one. Find its...

Question 19

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of
the inverse.
-$$f ( x ) = \sqrt { 5 + 4 x }$$

Accepted Answer

The domain of $f(x) = \sqrt{5 + 4x}$ is $x \geq -\frac{5}{4}$ because the expression under the square root must be non-negative. The range of $f(x)$ is $y \geq 0$ since square roots are always non-negative. For the inverse function, the domain and range swap roles, so the domain of $f^{-1}(x)$ is $x \geq 0$ and the range is $y \geq -\frac{5}{4}$.

Question 20

Find the inverse function of f. State the domain and range of f.
-$$f ( x ) = \frac { 3 x - 2 } { x + 5 }$$

Accepted Answer

To find the inverse function, swap $x$ and $y$ and solve for $y$. For $f(x) = \frac{3x - 2}{x + 5}$, setting $y = \frac{3x - 2}{x + 5}$ and solving for $x$ in terms of $y$ gives $x = \frac{5y + 2}{3 - y}$, which is the inverse function. The domain of $f$ excludes $x = -5$ to avoid division by zero, making the range of $f$ all real numbers except $y = \frac{2}{3}$, which is the horizontal asymptote of $f$, not directly related to the choices given.

Quiz 4: Exponential and Logarithmic Functions

Find a formula for the inverse of the function described below. - $32 ^ { \circ }$ Fahrenheit $= 0 ^ { \circ }$ Celsius. A function that converts temperatures in Celsius to those in Fahrenheit is $f ( x ) = \frac { 9 } { 5 } x + 32$

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - $f ( x ) = - 7 x - 4$

Write the word or phrase that best completes each statement or answers the question. Solve the problem. -The profit P for selling x items is given by the equation P(x) = 2x - 500. Express the sales amount x as a function of the profit P.

Approximate the value using a calculator. Express answer rounded to three decimal places. - $3 ^ { 3.3 }$

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - $f ( x ) = \frac { 3 } { x - 1 }$

Find a formula for the inverse of the function described below. -A size 44 dress in Country C is size 2 in Country D. A function that converts dress sizes in Country C to those in Country D is $f ( x ) = \frac { x } { 2 } - 20$

The function f is one-to-one. Find its inverse. - $f ( x ) = \sqrt [ 3 ] { x + 5 }$

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - $f ( x ) = \frac { 4 } { 6 x + 5 }$

The function f is one-to-one. Find its inverse. - $f ( x ) = \frac { 3 x - 6 } { 8 x - 8 }$

Find a formula for the inverse of the function described below. -An organization determines that the cost per person of chartering a bus is given by the formula $C ( x ) = \frac { 200 + 6 x } { x }$ where x is the number of people in the group and C(x) is in dollars.

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - $f ( x ) = \sqrt { 5 x + 4 }$

Find a formula for the inverse of the function described below. -A size 4 dress in Country C is size 40 in Country D. A function that converts dress sizes in Country C to those in Country D is f(x) = 2(x + 16) .

Find a formula for the inverse of the function described below. -A size 4 dress in Country $C$ is size 42 in Country D. A function that converts dress sizes in Country $C$ to those in Country $D$ is $f ( x ) = x + 38$ .

The function f is one-to-one. Find its inverse. - $f ( x ) = \sqrt { x - 1 }$

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - $f ( x ) = \sqrt { 5 + 4 x }$

Find the inverse function of f. State the domain and range of f. - $f ( x ) = \frac { 3 x - 2 } { x + 5 }$

Quiz 4: Exponential and Logarithmic Functions

Find a formula for the inverse of the function described below. - 32∘32 ^ { \circ }32∘ Fahrenheit =0∘= 0 ^ { \circ }=0∘ Celsius. A function that converts temperatures in Celsius to those in Fahrenheit is f(x) =95x+32f ( x ) = \frac { 9 } { 5 } x + 32f(x) =59​x+32

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - f(x) =−7x−4f ( x ) = - 7 x - 4f(x) =−7x−4

Write the word or phrase that best completes each statement or answers the question. Solve the problem. -The profit P for selling x items is given by the equation P(x) = 2x - 500. Express the sales amount x as a function of the profit P.

Approximate the value using a calculator. Express answer rounded to three decimal places. - 33.33 ^ { 3.3 }33.3

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - f(x) =3x−1f ( x ) = \frac { 3 } { x - 1 }f(x) =x−13​

Find a formula for the inverse of the function described below. -A size 44 dress in Country C is size 2 in Country D. A function that converts dress sizes in Country C to those in Country D is f(x) =x2−20f ( x ) = \frac { x } { 2 } - 20f(x) =2x​−20

The function f is one-to-one. Find its inverse. - f(x) =x+53f ( x ) = \sqrt [ 3 ] { x + 5 }f(x) =3x+5​

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - f(x) =46x+5f ( x ) = \frac { 4 } { 6 x + 5 }f(x) =6x+54​

The function f is one-to-one. Find its inverse. - f(x) =3x−68x−8f ( x ) = \frac { 3 x - 6 } { 8 x - 8 }f(x) =8x−83x−6​

Find a formula for the inverse of the function described below. -An organization determines that the cost per person of chartering a bus is given by the formula C(x) =200+6xxC ( x ) = \frac { 200 + 6 x } { x }C(x) =x200+6x​ where x is the number of people in the group and C(x) is in dollars.

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - f(x) =5x+4f ( x ) = \sqrt { 5 x + 4 }f(x) =5x+4​

Find a formula for the inverse of the function described below. -A size 4 dress in Country C is size 40 in Country D. A function that converts dress sizes in Country C to those in Country D is f(x) = 2(x + 16) .

Find a formula for the inverse of the function described below. -A size 4 dress in Country CCC is size 42 in Country D. A function that converts dress sizes in Country CCC to those in Country DDD is f(x) =x+38f ( x ) = x + 38f(x) =x+38 .

The function f is one-to-one. Find its inverse. - f(x) =x−1f ( x ) = \sqrt { x - 1 }f(x) =x−1​

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - f(x) =5+4xf ( x ) = \sqrt { 5 + 4 x }f(x) =5+4x​

Find the inverse function of f. State the domain and range of f. - f(x) =3x−2x+5f ( x ) = \frac { 3 x - 2 } { x + 5 }f(x) =x+53x−2​

Find a formula for the inverse of the function described below. - $32 ^ { \circ }$ Fahrenheit $= 0 ^ { \circ }$ Celsius. A function that converts temperatures in Celsius to those in Fahrenheit is $f ( x ) = \frac { 9 } { 5 } x + 32$

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - $f ( x ) = - 7 x - 4$

Approximate the value using a calculator. Express answer rounded to three decimal places. - $3 ^ { 3.3 }$

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - $f ( x ) = \frac { 3 } { x - 1 }$

Find a formula for the inverse of the function described below. -A size 44 dress in Country C is size 2 in Country D. A function that converts dress sizes in Country C to those in Country D is $f ( x ) = \frac { x } { 2 } - 20$

The function f is one-to-one. Find its inverse. - $f ( x ) = \sqrt [ 3 ] { x + 5 }$

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - $f ( x ) = \frac { 4 } { 6 x + 5 }$

The function f is one-to-one. Find its inverse. - $f ( x ) = \frac { 3 x - 6 } { 8 x - 8 }$

Find a formula for the inverse of the function described below. -An organization determines that the cost per person of chartering a bus is given by the formula $C ( x ) = \frac { 200 + 6 x } { x }$ where x is the number of people in the group and C(x) is in dollars.

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - $f ( x ) = \sqrt { 5 x + 4 }$

Find a formula for the inverse of the function described below. -A size 4 dress in Country $C$ is size 42 in Country D. A function that converts dress sizes in Country $C$ to those in Country $D$ is $f ( x ) = x + 38$ .

The function f is one-to-one. Find its inverse. - $f ( x ) = \sqrt { x - 1 }$

Determine i) the domain of the function, ii) the range of the function, iii) the domain of the inverse, and iv) the range of the inverse. - $f ( x ) = \sqrt { 5 + 4 x }$

Find the inverse function of f. State the domain and range of f. - $f ( x ) = \frac { 3 x - 2 } { x + 5 }$