Find the Inverse f(x)=7x^2+13

$f (x) = 7 x^{2} + 13$

Write $f (x) = 7 x^{2} + 13$ as an equation.

$y = 7 x^{2} + 13$

Interchange the variables.

$x = 7 y^{2} + 13$

Solve for $y$ .

Rewrite the equation as $7 y^{2} + 13 = x$ .

$7 y^{2} + 13 = x$

Subtract $13$ from both sides of the equation.

$7 y^{2} = x - 13$

Divide each term in $7 y^{2} = x - 13$ by $7$ and simplify.

Divide each term in $7 y^{2} = x - 13$ by $7$ .

$\frac{7 y^{2}}{7} = \frac{x}{7} + \frac{- 13}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 y^{2}}{7} = \frac{x}{7} + \frac{- 13}{7}$

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{x}{7} + \frac{- 13}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$y^{2} = \frac{x}{7} - \frac{13}{7}$

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

$y = \pm \sqrt{\frac{x}{7} - \frac{13}{7}}$

Simplify $\pm \sqrt{\frac{x}{7} - \frac{13}{7}}$ .

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x - 13}{7}}$

Rewrite $\sqrt{\frac{x - 13}{7}}$ as $\frac{\sqrt{x - 13}}{\sqrt{7}}$ .

$y = \pm \frac{\sqrt{x - 13}}{\sqrt{7}}$

Multiply $\frac{\sqrt{x - 13}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$y = \pm \frac{\sqrt{x - 13}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{x - 13}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$y = \pm \frac{\sqrt{x - 13} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Raise $\sqrt{7}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x - 13} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Raise $\sqrt{7}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x - 13} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

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

$y = \pm \frac{\sqrt{x - 13} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{x - 13} \sqrt{7}}{{\sqrt{7}}^{2}}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{x - 13} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

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

$y = \pm \frac{\sqrt{x - 13} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

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

$y = \pm \frac{\sqrt{x - 13} \sqrt{7}}{7^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \pm \frac{\sqrt{x - 13} \sqrt{7}}{7^{\frac{2}{2}}}$

Rewrite the expression.

$y = \pm \frac{\sqrt{x - 13} \sqrt{7}}{7^{1}}$

Evaluate the exponent.

$y = \pm \frac{\sqrt{x - 13} \sqrt{7}}{7}$

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(x - 13) \cdot 7}}{7}$

Reorder factors in $\pm \frac{\sqrt{(x - 13) \cdot 7}}{7}$ .

$y = \pm \frac{\sqrt{7 (x - 13)}}{7}$

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.

$y = \frac{\sqrt{7 (x - 13)}}{7}$

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

$y = - \frac{\sqrt{7 (x - 13)}}{7}$

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

$y = \frac{\sqrt{7 (x - 13)}}{7}$

$y = - \frac{\sqrt{7 (x - 13)}}{7}$

$y = \frac{\sqrt{7 (x - 13)}}{7}$

$y = - \frac{\sqrt{7 (x - 13)}}{7}$

$y = \frac{\sqrt{7 (x - 13)}}{7}$

$y = - \frac{\sqrt{7 (x - 13)}}{7}$

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

$f^{- 1} (x) = \frac{\sqrt{7 (x - 13)}}{7}, - \frac{\sqrt{7 (x - 13)}}{7}$

Verify if $f^{- 1} (x) = \frac{\sqrt{7 (x - 13)}}{7}, - \frac{\sqrt{7 (x - 13)}}{7}$ is the inverse of $f (x) = 7 x^{2} + 13$ .

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = 7 x^{2} + 13$ and $f^{- 1} (x) = \frac{\sqrt{7 (x - 13)}}{7}, - \frac{\sqrt{7 (x - 13)}}{7}$ and compare them.

Find the range of $f (x) = 7 x^{2} + 13$ .

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

Interval Notation:

$[13, \infty)$

Find the domain of $\frac{\sqrt{7 (x - 13)}}{7}$ .

Set the radicand in $\sqrt{7 (x - 13)}$ greater than or equal to $0$ to find where the expression is defined.

$7 (x - 13) \geq 0$

Solve for $x$ .

Divide each term in $7 (x - 13) \geq 0$ by $7$ and simplify.

Divide each term in $7 (x - 13) \geq 0$ by $7$ .

$\frac{7 (x - 13)}{7} \geq \frac{0}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 (x - 13)}{7} \geq \frac{0}{7}$

Divide $x - 13$ by $1$ .

$x - 13 \geq \frac{0}{7}$

Simplify the right side.

Divide $0$ by $7$ .

$x - 13 \geq 0$

Add $13$ to both sides of the inequality.

$x \geq 13$

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

$[13, \infty)$

Find the domain of $f (x) = 7 x^{2} + 13$ .

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 $f^{- 1} (x) = \frac{\sqrt{7 (x - 13)}}{7}, - \frac{\sqrt{7 (x - 13)}}{7}$ is the range of $f (x) = 7 x^{2} + 13$ and the range of $f^{- 1} (x) = \frac{\sqrt{7 (x - 13)}}{7}, - \frac{\sqrt{7 (x - 13)}}{7}$ is the domain of $f (x) = 7 x^{2} + 13$ , then $f^{- 1} (x) = \frac{\sqrt{7 (x - 13)}}{7}, - \frac{\sqrt{7 (x - 13)}}{7}$ is the inverse of $f (x) = 7 x^{2} + 13$ .

$f^{- 1} (x) = \frac{\sqrt{7 (x - 13)}}{7}, - \frac{\sqrt{7 (x - 13)}}{7}$

Find the Inverse f(x)=7x^2+13

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