Find the Inverse y=6x^2-5

$y = 6 x^{2} - 5$

Step 1

Interchange the variables.

$x = 6 y^{2} - 5$

Step 2

Solve for $y$ .

Rewrite the equation as $6 y^{2} - 5 = x$ .

$6 y^{2} - 5 = x$

Add $5$ to both sides of the equation.

$6 y^{2} = x + 5$

Divide each term in $6 y^{2} = x + 5$ by $6$ and simplify.

Divide each term in $6 y^{2} = x + 5$ by $6$ .

$\frac{6 y^{2}}{6} = \frac{x}{6} + \frac{5}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y^{2}}{6} = \frac{x}{6} + \frac{5}{6}$

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

$y^{2} = \frac{x}{6} + \frac{5}{6}$

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

$y = \pm \sqrt{\frac{x}{6} + \frac{5}{6}}$

Simplify $\pm \sqrt{\frac{x}{6} + \frac{5}{6}}$ .

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x + 5}{6}}$

Rewrite $\sqrt{\frac{x + 5}{6}}$ as $\frac{\sqrt{x + 5}}{\sqrt{6}}$ .

$y = \pm \frac{\sqrt{x + 5}}{\sqrt{6}}$

Multiply $\frac{\sqrt{x + 5}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$y = \pm \frac{\sqrt{x + 5}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{x + 5}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$y = \pm \frac{\sqrt{x + 5} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Raise $\sqrt{6}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x + 5} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Raise $\sqrt{6}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x + 5} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

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

$y = \pm \frac{\sqrt{x + 5} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{x + 5} \sqrt{6}}{{\sqrt{6}}^{2}}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{x + 5} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

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

$y = \pm \frac{\sqrt{x + 5} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

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

$y = \pm \frac{\sqrt{x + 5} \sqrt{6}}{6^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \pm \frac{\sqrt{x + 5} \sqrt{6}}{6^{\frac{2}{2}}}$

Rewrite the expression.

$y = \pm \frac{\sqrt{x + 5} \sqrt{6}}{6^{1}}$

Evaluate the exponent.

$y = \pm \frac{\sqrt{x + 5} \sqrt{6}}{6}$

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(x + 5) \cdot 6}}{6}$

Reorder factors in $\pm \frac{\sqrt{(x + 5) \cdot 6}}{6}$ .

$y = \pm \frac{\sqrt{6 (x + 5)}}{6}$

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.

$y = \frac{\sqrt{6 (x + 5)}}{6}$

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

$y = - \frac{\sqrt{6 (x + 5)}}{6}$

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

$y = \frac{\sqrt{6 (x + 5)}}{6}$

$y = - \frac{\sqrt{6 (x + 5)}}{6}$

$y = \frac{\sqrt{6 (x + 5)}}{6}$

$y = - \frac{\sqrt{6 (x + 5)}}{6}$

$y = \frac{\sqrt{6 (x + 5)}}{6}$

$y = - \frac{\sqrt{6 (x + 5)}}{6}$

Step 3

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

$f^{- 1} (x) = \frac{\sqrt{6 (x + 5)}}{6}, - \frac{\sqrt{6 (x + 5)}}{6}$

Step 4

Verify if $f^{- 1} (x) = \frac{\sqrt{6 (x + 5)}}{6}, - \frac{\sqrt{6 (x + 5)}}{6}$ is the inverse of $f (x) = 6 x^{2} - 5$ .

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = 6 x^{2} - 5$ and $f^{- 1} (x) = \frac{\sqrt{6 (x + 5)}}{6}, - \frac{\sqrt{6 (x + 5)}}{6}$ and compare them.

Find the range of $f (x) = 6 x^{2} - 5$ .

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

Interval Notation:

$[- 5, \infty)$

Find the domain of $\frac{\sqrt{6 (x + 5)}}{6}$ .

Set the radicand in $\sqrt{6 (x + 5)}$ greater than or equal to $0$ to find where the expression is defined.

$6 (x + 5) \geq 0$

Solve for $x$ .

Divide each term in $6 (x + 5) \geq 0$ by $6$ and simplify.

Divide each term in $6 (x + 5) \geq 0$ by $6$ .

$\frac{6 (x + 5)}{6} \geq \frac{0}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 (x + 5)}{6} \geq \frac{0}{6}$

Divide $x + 5$ by $1$ .

$x + 5 \geq \frac{0}{6}$

Simplify the right side.

Divide $0$ by $6$ .

$x + 5 \geq 0$

Subtract $5$ from both sides of the inequality.

$x \geq - 5$

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

$[- 5, \infty)$

Find the domain of $f (x) = 6 x^{2} - 5$ .

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 $f^{- 1} (x) = \frac{\sqrt{6 (x + 5)}}{6}, - \frac{\sqrt{6 (x + 5)}}{6}$ is the range of $f (x) = 6 x^{2} - 5$ and the range of $f^{- 1} (x) = \frac{\sqrt{6 (x + 5)}}{6}, - \frac{\sqrt{6 (x + 5)}}{6}$ is the domain of $f (x) = 6 x^{2} - 5$ , then $f^{- 1} (x) = \frac{\sqrt{6 (x + 5)}}{6}, - \frac{\sqrt{6 (x + 5)}}{6}$ is the inverse of $f (x) = 6 x^{2} - 5$ .

$f^{- 1} (x) = \frac{\sqrt{6 (x + 5)}}{6}, - \frac{\sqrt{6 (x + 5)}}{6}$

Step 5

Find the Inverse y=6x^2-5

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