Precalculus Examples

$f (x) = 3 x^{2} - 7$

Step 1

Write $f (x) = 3 x^{2} - 7$ as an equation.

$y = 3 x^{2} - 7$

Step 2

Interchange the variables.

$x = 3 y^{2} - 7$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $3 y^{2} - 7 = x$ .

$3 y^{2} - 7 = x$

Step 3.2

Add $7$ to both sides of the equation.

$3 y^{2} = x + 7$

Step 3.3

Divide each term in $3 y^{2} = x + 7$ by $3$ and simplify.

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Step 3.3.1

Divide each term in $3 y^{2} = x + 7$ by $3$ .

$\frac{3 y^{2}}{3} = \frac{x}{3} + \frac{7}{3}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{3 y^{2}}{3} = \frac{x}{3} + \frac{7}{3}$

Step 3.3.2.1.2

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

$y^{2} = \frac{x}{3} + \frac{7}{3}$

Step 3.4

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

$y = \pm \sqrt{\frac{x}{3} + \frac{7}{3}}$

Step 3.5

Simplify $\pm \sqrt{\frac{x}{3} + \frac{7}{3}}$ .

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Step 3.5.1

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x + 7}{3}}$

Step 3.5.2

Rewrite $\sqrt{\frac{x + 7}{3}}$ as $\frac{\sqrt{x + 7}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{x + 7}}{\sqrt{3}}$

Step 3.5.3

Multiply $\frac{\sqrt{x + 7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{x + 7}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 3.5.4

Combine and simplify the denominator.

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Step 3.5.4.1

Multiply $\frac{\sqrt{x + 7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{x + 7} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.5.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x + 7} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.5.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x + 7} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.5.4.4

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

$y = \pm \frac{\sqrt{x + 7} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.5.4.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{x + 7} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.5.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.5.4.6.1

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

$y = \pm \frac{\sqrt{x + 7} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.5.4.6.2

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

$y = \pm \frac{\sqrt{x + 7} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.5.4.6.3

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

$y = \pm \frac{\sqrt{x + 7} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.5.4.6.4

Cancel the common factor of $2$ .

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Step 3.5.4.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{x + 7} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.5.4.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{x + 7} \sqrt{3}}{3^{1}}$

Step 3.5.4.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{x + 7} \sqrt{3}}{3}$

Step 3.5.5

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(x + 7) \cdot 3}}{3}$

Step 3.5.6

Reorder factors in $\pm \frac{\sqrt{(x + 7) \cdot 3}}{3}$ .

$y = \pm \frac{\sqrt{3 (x + 7)}}{3}$

Step 3.6

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

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Step 3.6.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{\sqrt{3 (x + 7)}}{3}$

Step 3.6.2

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

$y = - \frac{\sqrt{3 (x + 7)}}{3}$

Step 3.6.3

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

$y = \frac{\sqrt{3 (x + 7)}}{3}$

$y = - \frac{\sqrt{3 (x + 7)}}{3}$

$y = \frac{\sqrt{3 (x + 7)}}{3}$

$y = - \frac{\sqrt{3 (x + 7)}}{3}$

$y = \frac{\sqrt{3 (x + 7)}}{3}$

$y = - \frac{\sqrt{3 (x + 7)}}{3}$

Step 4

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

$f^{- 1} (x) = \frac{\sqrt{3 (x + 7)}}{3}, - \frac{\sqrt{3 (x + 7)}}{3}$

Step 5

Verify if $f^{- 1} (x) = \frac{\sqrt{3 (x + 7)}}{3}, - \frac{\sqrt{3 (x + 7)}}{3}$ is the inverse of $f (x) = 3 x^{2} - 7$ .

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Step 5.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = 3 x^{2} - 7$ and $f^{- 1} (x) = \frac{\sqrt{3 (x + 7)}}{3}, - \frac{\sqrt{3 (x + 7)}}{3}$ and compare them.

Step 5.2

Find the range of $f (x) = 3 x^{2} - 7$ .

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Step 5.2.1

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

Interval Notation:

$[- 7, \infty)$

Step 5.3

Find the domain of $\frac{\sqrt{3 (x + 7)}}{3}$ .

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Step 5.3.1

Set the radicand in $\sqrt{3 (x + 7)}$ greater than or equal to $0$ to find where the expression is defined.

$3 (x + 7) \geq 0$

Step 5.3.2

Solve for $x$ .

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Step 5.3.2.1

Divide each term in $3 (x + 7) \geq 0$ by $3$ and simplify.

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Step 5.3.2.1.1

Divide each term in $3 (x + 7) \geq 0$ by $3$ .

$\frac{3 (x + 7)}{3} \geq \frac{0}{3}$

Step 5.3.2.1.2

Simplify the left side.

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Step 5.3.2.1.2.1

Cancel the common factor of $3$ .

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Step 5.3.2.1.2.1.1

Cancel the common factor.

$\frac{3 (x + 7)}{3} \geq \frac{0}{3}$

Step 5.3.2.1.2.1.2

Divide $x + 7$ by $1$ .

$x + 7 \geq \frac{0}{3}$

Step 5.3.2.1.3

Simplify the right side.

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Step 5.3.2.1.3.1

Divide $0$ by $3$ .

$x + 7 \geq 0$

Step 5.3.2.2

Subtract $7$ from both sides of the inequality.

$x \geq - 7$

Step 5.3.3

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

$[- 7, \infty)$

Step 5.4

Find the domain of $f (x) = 3 x^{2} - 7$ .

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Step 5.4.1

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.

$(- \infty, \infty)$

Step 5.5

Since the domain of $f^{- 1} (x) = \frac{\sqrt{3 (x + 7)}}{3}, - \frac{\sqrt{3 (x + 7)}}{3}$ is the range of $f (x) = 3 x^{2} - 7$ and the range of $f^{- 1} (x) = \frac{\sqrt{3 (x + 7)}}{3}, - \frac{\sqrt{3 (x + 7)}}{3}$ is the domain of $f (x) = 3 x^{2} - 7$ , then $f^{- 1} (x) = \frac{\sqrt{3 (x + 7)}}{3}, - \frac{\sqrt{3 (x + 7)}}{3}$ is the inverse of $f (x) = 3 x^{2} - 7$ .

$f^{- 1} (x) = \frac{\sqrt{3 (x + 7)}}{3}, - \frac{\sqrt{3 (x + 7)}}{3}$

Step 6