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7-86.

What is the inverse function of each of the following functions? Use any method to write an equation for the inverse function. Be sure to restrict domains as necessary. 7-86 HW eTool (Desmos)  Desmos Accessibility Homework Help ✎

1. $f\left(x\right) = x^{2} + 3$

This function squares the input then adds $3$. How can this be undone? Use the eTool to check your answer.

1. $f(x) = ( \frac { 1 } { 4 } x + 6 ) ^ { 3 }$

Let $f\left(x\right) = y$. Interchange $x$ and $y$. Solve for $y$ in the new equation.

$x = \left( \frac{1}{4}y+6\right)^3$

$\sqrt[3]{x}=\frac{1}{4}y+6$

$\sqrt[3]{x}-6=\frac{1}{4}y$

$f^{-1} = 4(\sqrt[3]{x}-6)$

1. $f(x) =\sqrt { 5 x - 6 }$

To undo square rooting, square both sides of the equation.

$f^{-1}(x)=\frac{x^2+6}{5}$

Use the eTool below to graph the function and its inverse.
Click the link at right for the full version of the eTool: Draw Inverse eTool
Domain restrictions can be added to a function by using brackets, for example $\text{{x>1}}$. Make sure both of the graphs are functions.