How do you find the inverse of f(x) = 5x^3 - 7?

May 16, 2018

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Inverse of the function color(red)(f(x)=5x^3-7 is given by color(blue)(root(3)((x+7)/5

Explanation:

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Note:

The Inverse of a function may NOT always be a function.

So, if the inverse of a function is a function by itself, then it is called an Inverse Function.

How do we determine these inverse relationships ?

Method 1

We can find the inverse of the function by simply swapping the ordered pairs.

Method 2

(a) Set the function to $y$

(b) Swap the $x$, $y$ variables

(c) Solve for $y$

Method 3

The graph of an inverse function is the reflection of the original graph over the line color(red)(y=x, called the Identity Line.

color(green)("Step 1 : "

Given the function : color(red)(y=f(x)=5x^3-7

Construct a data table for this function and graph it. Behavior of the Parent Graph shown color(green)("Step 2 : "

We use the Method 2 to solve

(a) Set the function to $y$

$y = 5 {x}^{3} - 7$

(b) Swap the $x$, $y$ variables

$x = 5 {y}^{3} - 7$

(c) Solve for $y$

We have, $x = 5 {y}^{3} - 7$

Subtract $5 {y}^{3}$ from both sides.

$\Rightarrow x - 5 {y}^{3} = 5 {y}^{3} - 7 - 5 {y}^{3}$

$\Rightarrow x - 5 {y}^{3} = \cancel{5 {y}^{3}} - 7 - \cancel{5 {y}^{3}}$

$\Rightarrow x - 5 {y}^{3} = - 7$

Subtract $x$ from both sides

$\Rightarrow x - 5 {y}^{3} - x = - 7 - x$

$\Rightarrow \cancel{x} - 5 {y}^{3} - \cancel{x} = - 7 - x$

Multiply both sides by $\left(- 1\right)$ to remove negative signs.

$\Rightarrow \left(- 1\right) \left(- 5 {y}^{3}\right) = \left(- 1\right) \left(- 7 - x\right)$

$\Rightarrow 5 {y}^{3} = 7 + x$

$\Rightarrow 5 {y}^{3} = x + 7$

Divide both sides by $5$

$\Rightarrow \frac{5 {y}^{3}}{5} = \frac{x + 7}{5}$

$\Rightarrow \frac{\cancel{5} {y}^{3}}{\cancel{5}} = \frac{x + 7}{5}$

${y}^{3} = \frac{x + 7}{5}$

Take Cube Root on both sides

$\Rightarrow \sqrt{{y}^{3}} = \sqrt{\frac{x + 7}{5}}$

Cube and the Cube Root cancel each other.

$\Rightarrow y = \sqrt{\frac{x + 7}{5}}$

Hence,

Inverse of the function color(red)(f(x)=5x^3-7 is given by color(blue)(root(3)((x+7)/5

color(green)("Step 3 : "

Explore the graph: 