Algebra Examples

$f (x) = - \frac{1}{2} \sqrt{x + 3}$ , $x \geq - 3$

Step 1

Find the range of the given function.

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Step 1.1

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

$(- \infty, 0]$

Step 1.2

Convert $(- \infty, 0]$ to an inequality.

$y \leq 0$

Step 2

Find the inverse.

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Step 2.1

Interchange the variables.

$x = - \frac{1}{2} \cdot \sqrt{y + 3}$

Step 2.2

Solve for $y$ .

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Step 2.2.1

Rewrite the equation as $- \frac{1}{2} \cdot \sqrt{y + 3} = x$ .

$- \frac{1}{2} \cdot \sqrt{y + 3} = x$

Step 2.2.2

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{1}{2} \cdot \sqrt{y + 3}) = - 2 x$

Step 2.2.3

Simplify the left side.

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Step 2.2.3.1

Simplify $- 2 (- \frac{1}{2} \cdot \sqrt{y + 3})$ .

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Step 2.2.3.1.1

Combine $\sqrt{y + 3}$ and $\frac{1}{2}$ .

$- 2 (- \frac{\sqrt{y + 3}}{2}) = - 2 x$

Step 2.2.3.1.2

Cancel the common factor of $2$ .

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Step 2.2.3.1.2.1

Move the leading negative in $- \frac{\sqrt{y + 3}}{2}$ into the numerator.

$- 2 \frac{- \sqrt{y + 3}}{2} = - 2 x$

Step 2.2.3.1.2.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- \sqrt{y + 3}}{2} = - 2 x$

Step 2.2.3.1.2.3

Cancel the common factor.

$2 \cdot - 1 \frac{- \sqrt{y + 3}}{2} = - 2 x$

Step 2.2.3.1.2.4

Rewrite the expression.

$- - \sqrt{y + 3} = - 2 x$

Step 2.2.3.1.3

Multiply.

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Step 2.2.3.1.3.1

Multiply $- 1$ by $- 1$ .

$1 \sqrt{y + 3} = - 2 x$

Step 2.2.3.1.3.2

Multiply $\sqrt{y + 3}$ by $1$ .

$\sqrt{y + 3} = - 2 x$

Step 2.2.4

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{y + 3}}^{2} = {(- 2 x)}^{2}$

Step 2.2.5

Simplify each side of the equation.

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Step 2.2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y + 3}$ as ${(y + 3)}^{\frac{1}{2}}$ .

${({(y + 3)}^{\frac{1}{2}})}^{2} = {(- 2 x)}^{2}$

Step 2.2.5.2

Simplify the left side.

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Step 2.2.5.2.1

Simplify ${({(y + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.5.2.1.1

Multiply the exponents in ${({(y + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.5.2.1.1.1

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

${(y + 3)}^{\frac{1}{2} \cdot 2} = {(- 2 x)}^{2}$

Step 2.2.5.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.5.2.1.1.2.1

Cancel the common factor.

${(y + 3)}^{\frac{1}{2} \cdot 2} = {(- 2 x)}^{2}$

Step 2.2.5.2.1.1.2.2

Rewrite the expression.

${(y + 3)}^{1} = {(- 2 x)}^{2}$

Step 2.2.5.2.1.2

Simplify.

$y + 3 = {(- 2 x)}^{2}$

Step 2.2.5.3

Simplify the right side.

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Step 2.2.5.3.1

Simplify ${(- 2 x)}^{2}$ .

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Step 2.2.5.3.1.1

Apply the product rule to $- 2 x$ .

$y + 3 = {(- 2)}^{2} x^{2}$

Step 2.2.5.3.1.2

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

$y + 3 = 4 x^{2}$

Step 2.2.6

Subtract $3$ from both sides of the equation.

$y = 4 x^{2} - 3$

Step 2.3

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

$f^{- 1} (x) = 4 x^{2} - 3$

Step 3

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

$f^{- 1} (x) = 4 x^{2} - 3, x \leq 0$

Step 4