5.7 Inverses and Radical Functions

Given function, find the inverse function.

Find the inverse of the function 0,3,1,5,2,8,3,4. Determine the domain and range of the inverse function.

Find the domain of a radical function.

Find the domain of the function $f \left(x\right) = \sqrt{3 x - 4}$ . Write the domain in interval notation.

Show that $f \left(x\right) = \frac{1}{x + 1}$ and $f^{- 1} \left(x\right) = \frac{1}{x} - 1$ are inverses, for $x \neq 0 , −1$ .

Show that $f \left(x\right) = \frac{x + 5}{3}$ and $f^{- 1} \left(x\right) = 3 x - 5$ are inverses.

Find the inverse of the function $f \left(x\right) = 5 x^{3} + 1.$

Find the inverse function of $f \left(x\right) = \sqrt[3]{x + 4} .$

Restrict the domain and then find the inverse of

Find the inverse of the function $f \left(x\right) = x^{2} + 1 ,$ on the domain $x \geq 0.$

Restrict the domain of the function $f \left(x\right) = \sqrt{x - 4}$ and then find the inverse.

Restrict the domain and then find the inverse of the function $f \left(x\right) = \sqrt{2 x + 3} .$

Park rangers construct a mound of gravel in the shape of a cone with the height equal to twice the radius. The volume of the cone in terms of the radius is given by

Find the inverse of the function $V = \frac{2}{3} \pi r^{3}$ that determines the volume $V$ of a cone and is a function of the radius $r .$ Then use the inverse function to calculate the radius of such a mound of gravel measuring 100 cubic feet. Use $\pi = 3.14.$

The function $C = \frac{20 + 0.4 n}{100 + n}$ represents the concentration $C$ of an acid solution after $n$ mL of 40% solution has been added to 100 mL of a 20% solution. First, find the inverse of the function; that is, find an expression for $n$ in terms of $C .$ Then use your result to determine how much of the 40% solution should be added so that the final mixture is a 35% solution.

Find the inverse of the function $f \left(x\right) = \frac{x + 3}{x - 2} .$

Explain why we cannot find inverse functions for all polynomial functions.

When finding the inverse of a radical function, what restriction will we need to make?

Find the inverse of the function on the given domain.

$f \left(x\right) = \left(x - 4\right)^{2} , \left[4 , \infty\right)$

Find the inverse of the function on the given domain.

$f \left(x\right) = \left(x + 1\right)^{2} - 3 , \left[−1 , \infty\right)$

Find the inverse of the function on the given domain.

$f \left(x\right) = 12 - x^{2} , \left[0 , \infty\right)$

Find the inverse of the function on the given domain.

$f \left(x\right) = 2 x^{2} + 4 , \left[0 , \infty\right)$

Find the inverse of the functions.

$f \left(x\right) = 3 x^{3} + 1$

Find the inverse of the functions.

$f \left(x\right) = 4 - 2 x^{3}$

Find the inverse of the functions.

$f \left(x\right) = \sqrt{3 - 4 x}$

Find the inverse of the functions.

$f \left(x\right) = \sqrt{6 x - 8} + 5$

Find the inverse of the functions.

$f \left(x\right) = 3 - \sqrt[3]{x}$

Find the inverse of the functions.

$f \left(x\right) = \frac{3}{x - 4}$

Find the inverse of the functions.

$f \left(x\right) = \frac{x - 2}{x + 7}$

Find the inverse of the functions.

$f \left(x\right) = \frac{5 x + 1}{2 - 5 x}$

Find the inverse of the functions.

$f \left(x\right) = x^{2} + 4 x + 1 , \left[−2 , \infty\right)$

Find the inverse of the functions with $a , b , c$ positive real numbers.

$f \left(x\right) = a x^{3} + b$

Find the inverse of the functions with $a , b , c$ positive real numbers.

$f \left(x\right) = \sqrt{a x^{2} - b}$

Find the inverse of the functions with $a , b , c$ positive real numbers.

$f \left(x\right) = \frac{a x + b}{x + c}$

Determine the function described and then use it to answer the question.

An object dropped from a height of 600 feet has a height, $h \left(t\right) ,$ in feet after $t$ seconds have elapsed, such that $h \left(t\right) = 600 - 16 t^{2} .$ Express $t$ as a function of height $h ,$ and find the time to reach a height of 400 feet.

Determine the function described and then use it to answer the question.

The surface area, $A ,$ of a sphere in terms of its radius, $r ,$ is given by $A \left(r\right) = 4 \pi r^{2} .$ Express $r$ as a function of $A ,$ and find the radius of a sphere with a surface area of 1000 square inches.

Determine the function described and then use it to answer the question.

The period $T ,$ in seconds, of a simple pendulum as a function of its length $l ,$ in feet, is given by $T \left(l\right) = 2 \pi \sqrt{\frac{l}{32.2}}$ . Express $l$ as a function of $T$ and determine the length of a pendulum with period of 2 seconds.

Determine the function described and then use it to answer the question.

The surface area, $A ,$ of a cylinder in terms of its radius, $r ,$ and height, $h ,$ is given by $A = 2 \pi r^{2} + 2 \pi r h .$ If the height of the cylinder is 4 feet, express the radius as a function of $A$ and find the radius if the surface area is 200 square feet.

Determine the function described and then use it to answer the question.

Consider a cone with height of 30 feet. Express the radius, $r ,$ in terms of the volume, $V ,$ and find the radius of a cone with volume of 1000 cubic feet.

Find the inverse of the function on the given domain.

$f \left(x\right) = 3 x^{2} + 5 , \left(- \infty , 0\right]$

Find the inverse of the function on the given domain.

$f \left(x\right) = 9 - x^{2} , \left[0 , \infty\right)$

Find the inverse of the functions.

$f \left(x\right) = x^{3} + 5$

Find the inverse of the functions.

$f \left(x\right) = 4 - x^{3}$

Find the inverse of the functions.

$f \left(x\right) = \sqrt{2 x + 1}$

Find the inverse of the functions.

$f \left(x\right) = 9 + \sqrt{4 x - 4}$

Find the inverse of the functions.

$f \left(x\right) = 9 + 2 \sqrt[3]{x}$

Find the inverse of the functions.

$f \left(x\right) = \frac{2}{x + 8}$

Find the inverse of the functions.

$f \left(x\right) = \frac{x + 3}{x + 7}$

Find the inverse of the functions.

$f \left(x\right) = \frac{3 x + 4}{5 - 4 x}$

Find the inverse of the functions.

$f \left(x\right) = x^{2} + 2 x , \left[- 1 , \infty\right)$

Find the inverse of the functions.

$f \left(x\right) = x^{2} - 6 x + 3 , \left[3 , \infty\right)$

Find the inverse of the functions with $a , b , c$ positive real numbers.

$f \left(x\right) = x^{2} + b x$

Find the inverse of the functions with $a , b , c$ positive real numbers.

$f \left(x\right) = \sqrt[3]{a x + b}$

Determine the function described and then use it to answer the question.

An object dropped from a height of 200 meters has a height, $h \left(t\right) ,$ in meters after $t$ seconds have lapsed, such that $h \left(t\right) = 200 - 4.9 t^{2} .$ Express $t$ as a function of height, $h ,$ and find the time to reach a height of 50 meters.

Determine the function described and then use it to answer the question.

The volume, $V ,$ of a sphere in terms of its radius, $r ,$ is given by $V \left(r\right) = \frac{4}{3} \pi r^{3} .$ Express $r$ as a function of $V ,$ and find the radius of a sphere with volume of 200 cubic feet.

Determine the function described and then use it to answer the question.

A container holds 100 ml of a solution that is 25 ml acid. If $n$ ml of a solution that is 60% acid is added, the function $C \left(n\right) = \frac{25 + .6 n}{100 + n}$ gives the concentration, $C ,$ as a function of the number of ml added, $n .$ Express $n$ as a function of $C$ and determine the number of mL that need to be added to have a solution that is 50% acid.

Determine the function described and then use it to answer the question.

The volume of a cylinder , $V ,$ in terms of radius, $r ,$ and height, $h ,$ is given by $V = \pi r^{2} h .$ If a cylinder has a height of 6 meters, express the radius as a function of $V$ and find the radius of a cylinder with volume of 300 cubic meters.

Determine the function described and then use it to answer the question.

The volume of a right circular cone, $V ,$ in terms of its radius, $r ,$ and its height, $h ,$ is given by $V = \frac{1}{3} \pi r^{2} h .$ Express $r$ in terms of $V$ if the height of the cone is 12 inches and find the radius of a cone with volume of 50 cubic inches.