Find the Inverse f(x)=2(x-4)^2

$f (x) = 2 {(x - 4)}^{2}$

Step 1

Write $f (x) = 2 {(x - 4)}^{2}$ as an equation.

$y = 2 {(x - 4)}^{2}$

Step 2

Interchange the variables.

$x = 2 {(y - 4)}^{2}$

Step 3

Solve for $y$ .

Rewrite the equation as $2 {(y - 4)}^{2} = x$ .

$2 {(y - 4)}^{2} = x$

Divide each term in $2 {(y - 4)}^{2} = x$ by $2$ and simplify.

Divide each term in $2 {(y - 4)}^{2} = x$ by $2$ .

$\frac{2 {(y - 4)}^{2}}{2} = \frac{x}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 {(y - 4)}^{2}}{2} = \frac{x}{2}$

Divide ${(y - 4)}^{2}$ by $1$ .

${(y - 4)}^{2} = \frac{x}{2}$

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

$y - 4 = \pm \sqrt{\frac{x}{2}}$

Simplify $\pm \sqrt{\frac{x}{2}}$ .

Rewrite $\sqrt{\frac{x}{2}}$ as $\frac{\sqrt{x}}{\sqrt{2}}$ .

$y - 4 = \pm \frac{\sqrt{x}}{\sqrt{2}}$

Multiply $\frac{\sqrt{x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y - 4 = \pm \frac{\sqrt{x}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y - 4 = \pm \frac{\sqrt{x} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$y - 4 = \pm \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$y - 4 = \pm \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y - 4 = \pm \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$y - 4 = \pm \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$y - 4 = \pm \frac{\sqrt{x} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y - 4 = \pm \frac{\sqrt{x} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$y - 4 = \pm \frac{\sqrt{x} \sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y - 4 = \pm \frac{\sqrt{x} \sqrt{2}}{2^{\frac{2}{2}}}$

Rewrite the expression.

$y - 4 = \pm \frac{\sqrt{x} \sqrt{2}}{2^{1}}$

Evaluate the exponent.

$y - 4 = \pm \frac{\sqrt{x} \sqrt{2}}{2}$

Combine using the product rule for radicals.

$y - 4 = \pm \frac{\sqrt{x \cdot 2}}{2}$

Reorder factors in $\pm \frac{\sqrt{x \cdot 2}}{2}$ .

$y - 4 = \pm \frac{\sqrt{2 x}}{2}$

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.

$y - 4 = \frac{\sqrt{2 x}}{2}$

Add $4$ to both sides of the equation.

$y = \frac{\sqrt{2 x}}{2} + 4$

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

$y - 4 = - \frac{\sqrt{2 x}}{2}$

Add $4$ to both sides of the equation.

$y = - \frac{\sqrt{2 x}}{2} + 4$

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

$y = \frac{\sqrt{2 x}}{2} + 4$

$y = - \frac{\sqrt{2 x}}{2} + 4$

$y = \frac{\sqrt{2 x}}{2} + 4$

$y = - \frac{\sqrt{2 x}}{2} + 4$

$y = \frac{\sqrt{2 x}}{2} + 4$

$y = - \frac{\sqrt{2 x}}{2} + 4$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt{2 x}}{2} + 4, - \frac{\sqrt{2 x}}{2} + 4$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt{2 x}}{2} + 4, - \frac{\sqrt{2 x}}{2} + 4$ is the inverse of $f (x) = 2 {(x - 4)}^{2}$ .

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = 2 {(x - 4)}^{2}$ and $f^{- 1} (x) = \frac{\sqrt{2 x}}{2} + 4, - \frac{\sqrt{2 x}}{2} + 4$ and compare them.

Find the range of $f (x) = 2 {(x - 4)}^{2}$ .

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

Interval Notation:

$[0, \infty)$

Find the domain of $\frac{\sqrt{2 x}}{2} + 4$ .

Set the radicand in $\sqrt{2 x}$ greater than or equal to $0$ to find where the expression is defined.

$2 x \geq 0$

Divide each term in $2 x \geq 0$ by $2$ and simplify.

Divide each term in $2 x \geq 0$ by $2$ .

$\frac{2 x}{2} \geq \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{0}{2}$

Divide $x$ by $1$ .

$x \geq \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x \geq 0$

The domain is all values of $x$ that make the expression defined.

$[0, \infty)$

Find the domain of $f (x) = 2 {(x - 4)}^{2}$ .

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

$(- \infty, \infty)$

Since the domain of $f^{- 1} (x) = \frac{\sqrt{2 x}}{2} + 4, - \frac{\sqrt{2 x}}{2} + 4$ is the range of $f (x) = 2 {(x - 4)}^{2}$ and the range of $f^{- 1} (x) = \frac{\sqrt{2 x}}{2} + 4, - \frac{\sqrt{2 x}}{2} + 4$ is the domain of $f (x) = 2 {(x - 4)}^{2}$ , then $f^{- 1} (x) = \frac{\sqrt{2 x}}{2} + 4, - \frac{\sqrt{2 x}}{2} + 4$ is the inverse of $f (x) = 2 {(x - 4)}^{2}$ .

$f^{- 1} (x) = \frac{\sqrt{2 x}}{2} + 4, - \frac{\sqrt{2 x}}{2} + 4$

Step 6

Find the Inverse f(x)=2(x-4)^2

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