Find the Inverse y=x^2+12x

$y = x^{2} + 12 x$

Step 1

Interchange the variables.

$x = y^{2} + 12 y$

Step 2

Solve for $y$ .

Rewrite the equation as $y^{2} + 12 y = x$ .

$y^{2} + 12 y = x$

Subtract $x$ from both sides of the equation.

$y^{2} + 12 y - x = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = 12$ , and $c = - x$ into the quadratic formula and solve for $y$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (1 \cdot (- x))}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot 1 \cdot (- 1 x)}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 1$ .

Multiply $- 4$ by $1$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot (- 1 x)}}{2 \cdot 1}$

Multiply $- 4$ by $- 1$ .

$y = \frac{- 12 \pm \sqrt{144 + 4 x}}{2 \cdot 1}$

Factor $4$ out of $144 + 4 x$ .

Factor $4$ out of $144$ .

$y = \frac{- 12 \pm \sqrt{4 \cdot 36 + 4 x}}{2 \cdot 1}$

Factor $4$ out of $4 \cdot 36 + 4 x$ .

$y = \frac{- 12 \pm \sqrt{4 (36 + x)}}{2 \cdot 1}$

Rewrite $4 (36 + x)$ as $2^{2} (6^{2} + x)$ .

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 12 \pm \sqrt{2^{2} (36 + x)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$y = \frac{- 12 \pm \sqrt{2^{2} (6^{2} + x)}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{- 12 \pm 2 \sqrt{6^{2} + x}}{2 \cdot 1}$

Raise $6$ to the power of $2$ .

$y = \frac{- 12 \pm 2 \sqrt{36 + x}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{- 12 \pm 2 \sqrt{36 + x}}{2}$

Simplify $\frac{- 12 \pm 2 \sqrt{36 + x}}{2}$ .

$y = - 6 \pm \sqrt{36 + x}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot 1 \cdot (- 1 x)}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 1$ .

Multiply $- 4$ by $1$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot (- 1 x)}}{2 \cdot 1}$

Multiply $- 4$ by $- 1$ .

$y = \frac{- 12 \pm \sqrt{144 + 4 x}}{2 \cdot 1}$

Factor $4$ out of $144 + 4 x$ .

Factor $4$ out of $144$ .

$y = \frac{- 12 \pm \sqrt{4 \cdot 36 + 4 x}}{2 \cdot 1}$

Factor $4$ out of $4 \cdot 36 + 4 x$ .

$y = \frac{- 12 \pm \sqrt{4 (36 + x)}}{2 \cdot 1}$

Rewrite $4 (36 + x)$ as $2^{2} (6^{2} + x)$ .

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 12 \pm \sqrt{2^{2} (36 + x)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$y = \frac{- 12 \pm \sqrt{2^{2} (6^{2} + x)}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{- 12 \pm 2 \sqrt{6^{2} + x}}{2 \cdot 1}$

Raise $6$ to the power of $2$ .

$y = \frac{- 12 \pm 2 \sqrt{36 + x}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{- 12 \pm 2 \sqrt{36 + x}}{2}$

Simplify $\frac{- 12 \pm 2 \sqrt{36 + x}}{2}$ .

$y = - 6 \pm \sqrt{36 + x}$

Change the $\pm$ to $+$ .

$y = - 6 + \sqrt{36 + x}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot 1 \cdot (- 1 x)}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 1$ .

Multiply $- 4$ by $1$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot (- 1 x)}}{2 \cdot 1}$

Multiply $- 4$ by $- 1$ .

$y = \frac{- 12 \pm \sqrt{144 + 4 x}}{2 \cdot 1}$

Factor $4$ out of $144 + 4 x$ .

Factor $4$ out of $144$ .

$y = \frac{- 12 \pm \sqrt{4 \cdot 36 + 4 x}}{2 \cdot 1}$

Factor $4$ out of $4 \cdot 36 + 4 x$ .

$y = \frac{- 12 \pm \sqrt{4 (36 + x)}}{2 \cdot 1}$

Rewrite $4 (36 + x)$ as $2^{2} (6^{2} + x)$ .

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 12 \pm \sqrt{2^{2} (36 + x)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$y = \frac{- 12 \pm \sqrt{2^{2} (6^{2} + x)}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{- 12 \pm 2 \sqrt{6^{2} + x}}{2 \cdot 1}$

Raise $6$ to the power of $2$ .

$y = \frac{- 12 \pm 2 \sqrt{36 + x}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{- 12 \pm 2 \sqrt{36 + x}}{2}$

Simplify $\frac{- 12 \pm 2 \sqrt{36 + x}}{2}$ .

$y = - 6 \pm \sqrt{36 + x}$

Change the $\pm$ to $-$ .

$y = - 6 - \sqrt{36 + x}$

The final answer is the combination of both solutions.

$y = - 6 + \sqrt{36 + x}$

$y = - 6 - \sqrt{36 + x}$

$y = - 6 + \sqrt{36 + x}$

$y = - 6 - \sqrt{36 + x}$

Step 3

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

$f^{- 1} (x) = - 6 + \sqrt{36 + x}, - 6 - \sqrt{36 + x}$

Step 4

Verify if $f^{- 1} (x) = - 6 + \sqrt{36 + x}, - 6 - \sqrt{36 + x}$ is the inverse of $f (x) = x^{2} + 12 x$ .

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = x^{2} + 12 x$ and $f^{- 1} (x) = - 6 + \sqrt{36 + x}, - 6 - \sqrt{36 + x}$ and compare them.

Find the range of $f (x) = x^{2} + 12 x$ .

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

Interval Notation:

$[- 36, \infty)$

Find the domain of $- 6 + \sqrt{36 + x}$ .

Set the radicand in $\sqrt{36 + x}$ greater than or equal to $0$ to find where the expression is defined.

$36 + x \geq 0$

Subtract $36$ from both sides of the inequality.

$x \geq - 36$

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

$[- 36, \infty)$

Find the domain of $f (x) = x^{2} + 12 x$ .

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 $f^{- 1} (x) = - 6 + \sqrt{36 + x}, - 6 - \sqrt{36 + x}$ is the range of $f (x) = x^{2} + 12 x$ and the range of $f^{- 1} (x) = - 6 + \sqrt{36 + x}, - 6 - \sqrt{36 + x}$ is the domain of $f (x) = x^{2} + 12 x$ , then $f^{- 1} (x) = - 6 + \sqrt{36 + x}, - 6 - \sqrt{36 + x}$ is the inverse of $f (x) = x^{2} + 12 x$ .

$f^{- 1} (x) = - 6 + \sqrt{36 + x}, - 6 - \sqrt{36 + x}$

Step 5

Find the Inverse y=x^2+12x

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