Find the Inverse f(x)=1/7x^2 , x>=0

$f (x) = \frac{1}{7} x^{2}$ , $x \geq 0$

Step 1

Find the range of the given function.

The range is the set of all valid $y$ values. Use the graph to find the range.

$[0, \infty)$

Convert $[0, \infty)$ to an inequality.

$y \geq 0$

Step 2

Find the inverse.

Interchange the variables.

$x = \frac{1}{7} y^{2}$

Solve for $y$ .

Rewrite the equation as $\frac{1}{7} y^{2} = x$ .

$\frac{1}{7} y^{2} = x$

Multiply both sides of the equation by $7$ .

$7 (\frac{1}{7} y^{2}) = 7 x$

Simplify the left side.

Simplify $7 (\frac{1}{7} y^{2})$ .

Combine $\frac{1}{7}$ and $y^{2}$ .

$7 \frac{y^{2}}{7} = 7 x$

Cancel the common factor of $7$ .

Cancel the common factor.

$7 \frac{y^{2}}{7} = 7 x$

Rewrite the expression.

$y^{2} = 7 x$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{7 x}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{7 x}$

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{7 x}$

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{7 x}$

$y = - \sqrt{7 x}$

$y = \sqrt{7 x}$

$y = - \sqrt{7 x}$

$y = \sqrt{7 x}$

$y = - \sqrt{7 x}$

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \sqrt{7 x}, - \sqrt{7 x}$

Step 3

Find the inverse using the domain and the range of the original function.

$f^{- 1} (x) = \sqrt{7 x}, x \geq 0$

Step 4