Precalculus Examples

$f (x) = \sqrt{3 x - 1} + 5$

Step 1

Write $f (x) = \sqrt{3 x - 1} + 5$ as an equation.

$y = \sqrt{3 x - 1} + 5$

Step 2

Interchange the variables.

$x = \sqrt{3 y - 1} + 5$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt{3 y - 1} + 5 = x$ .

$\sqrt{3 y - 1} + 5 = x$

Step 3.2

Subtract $5$ from both sides of the equation.

$\sqrt{3 y - 1} = x - 5$

Step 3.3

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

${\sqrt{3 y - 1}}^{2} = {(x - 5)}^{2}$

Step 3.4

Simplify each side of the equation.

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Step 3.4.1

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

${({(3 y - 1)}^{\frac{1}{2}})}^{2} = {(x - 5)}^{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

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

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Step 3.4.2.1.1

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

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Step 3.4.2.1.1.1

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

${(3 y - 1)}^{\frac{1}{2} \cdot 2} = {(x - 5)}^{2}$

Step 3.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.2.1.1.2.1

Cancel the common factor.

${(3 y - 1)}^{\frac{1}{2} \cdot 2} = {(x - 5)}^{2}$

Step 3.4.2.1.1.2.2

Rewrite the expression.

${(3 y - 1)}^{1} = {(x - 5)}^{2}$

Step 3.4.2.1.2

Simplify.

$3 y - 1 = {(x - 5)}^{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(x - 5)}^{2}$ .

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Step 3.4.3.1.1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$3 y - 1 = (x - 5) (x - 5)$

Step 3.4.3.1.2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 3.4.3.1.2.1

Apply the distributive property.

$3 y - 1 = x (x - 5) - 5 (x - 5)$

Step 3.4.3.1.2.2

Apply the distributive property.

$3 y - 1 = x \cdot x + x \cdot - 5 - 5 (x - 5)$

Step 3.4.3.1.2.3

Apply the distributive property.

$3 y - 1 = x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5$

Step 3.4.3.1.3

Simplify and combine like terms.

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Step 3.4.3.1.3.1

Simplify each term.

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Step 3.4.3.1.3.1.1

Multiply $x$ by $x$ .

$3 y - 1 = x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5$

Step 3.4.3.1.3.1.2

Move $- 5$ to the left of $x$ .

$3 y - 1 = x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5$

Step 3.4.3.1.3.1.3

Multiply $- 5$ by $- 5$ .

$3 y - 1 = x^{2} - 5 x - 5 x + 25$

Step 3.4.3.1.3.2

Subtract $5 x$ from $- 5 x$ .

$3 y - 1 = x^{2} - 10 x + 25$

Step 3.5

Solve for $y$ .

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Step 3.5.1

Move all terms not containing $y$ to the right side of the equation.

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Step 3.5.1.1

Add $1$ to both sides of the equation.

$3 y = x^{2} - 10 x + 25 + 1$

Step 3.5.1.2

Add $25$ and $1$ .

$3 y = x^{2} - 10 x + 26$

Step 3.5.2

Divide each term in $3 y = x^{2} - 10 x + 26$ by $3$ and simplify.

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Step 3.5.2.1

Divide each term in $3 y = x^{2} - 10 x + 26$ by $3$ .

$\frac{3 y}{3} = \frac{x^{2}}{3} + \frac{- 10 x}{3} + \frac{26}{3}$

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Cancel the common factor of $3$ .

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Step 3.5.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{x^{2}}{3} + \frac{- 10 x}{3} + \frac{26}{3}$

Step 3.5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{3} + \frac{- 10 x}{3} + \frac{26}{3}$

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Move the negative in front of the fraction.

$y = \frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}$

Step 4

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

$f^{- 1} (x) = \frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}$

Step 5

Verify if $f^{- 1} (x) = \frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}$ is the inverse of $f (x) = \sqrt{3 x - 1} + 5$ .

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate $f^{- 1} (f (x))$ .

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Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate $f^{- 1} (\sqrt{3 x - 1} + 5)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{{(\sqrt{3 x - 1} + 5)}^{2}}{3} - \frac{10 (\sqrt{3 x - 1} + 5)}{3} + \frac{26}{3}$

Step 5.2.3

Combine the numerators over the common denominator.

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{{(\sqrt{3 x - 1} + 5)}^{2} - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Rewrite ${(\sqrt{3 x - 1} + 5)}^{2}$ as $(\sqrt{3 x - 1} + 5) (\sqrt{3 x - 1} + 5)$ .

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{(\sqrt{3 x - 1} + 5) (\sqrt{3 x - 1} + 5) - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.2

Expand $(\sqrt{3 x - 1} + 5) (\sqrt{3 x - 1} + 5)$ using the FOIL Method.

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Step 5.2.4.2.1

Apply the distributive property.

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{\sqrt{3 x - 1} (\sqrt{3 x - 1} + 5) + 5 (\sqrt{3 x - 1} + 5) - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.2.2

Apply the distributive property.

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{\sqrt{3 x - 1} \sqrt{3 x - 1} + \sqrt{3 x - 1} \cdot 5 + 5 (\sqrt{3 x - 1} + 5) - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.2.3

Apply the distributive property.

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{\sqrt{3 x - 1} \sqrt{3 x - 1} + \sqrt{3 x - 1} \cdot 5 + 5 \sqrt{3 x - 1} + 5 \cdot 5 - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.3

Simplify and combine like terms.

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Step 5.2.4.3.1

Simplify each term.

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Step 5.2.4.3.1.1

Multiply $\sqrt{3 x - 1} \sqrt{3 x - 1}$ .

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Step 5.2.4.3.1.1.1

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

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{\sqrt{3 x - 1} \sqrt{3 x - 1} + \sqrt{3 x - 1} \cdot 5 + 5 \sqrt{3 x - 1} + 5 \cdot 5 - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.3.1.1.2

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

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{\sqrt{3 x - 1} \sqrt{3 x - 1} + \sqrt{3 x - 1} \cdot 5 + 5 \sqrt{3 x - 1} + 5 \cdot 5 - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.3.1.1.3

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

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{{\sqrt{3 x - 1}}^{1 + 1} + \sqrt{3 x - 1} \cdot 5 + 5 \sqrt{3 x - 1} + 5 \cdot 5 - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.3.1.1.4

Add $1$ and $1$ .

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{{\sqrt{3 x - 1}}^{2} + \sqrt{3 x - 1} \cdot 5 + 5 \sqrt{3 x - 1} + 5 \cdot 5 - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.3.1.2

Rewrite ${\sqrt{3 x - 1}}^{2}$ as $3 x - 1$ .

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Step 5.2.4.3.1.2.1

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

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{{({(3 x - 1)}^{\frac{1}{2}})}^{2} + \sqrt{3 x - 1} \cdot 5 + 5 \sqrt{3 x - 1} + 5 \cdot 5 - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.3.1.2.2

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

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{{(3 x - 1)}^{\frac{1}{2} \cdot 2} + \sqrt{3 x - 1} \cdot 5 + 5 \sqrt{3 x - 1} + 5 \cdot 5 - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.3.1.2.3

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

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{{(3 x - 1)}^{\frac{2}{2}} + \sqrt{3 x - 1} \cdot 5 + 5 \sqrt{3 x - 1} + 5 \cdot 5 - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.3.1.2.4

Cancel the common factor of $2$ .

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Step 5.2.4.3.1.2.4.1

Cancel the common factor.

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{{(3 x - 1)}^{\frac{2}{2}} + \sqrt{3 x - 1} \cdot 5 + 5 \sqrt{3 x - 1} + 5 \cdot 5 - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.3.1.2.4.2

Rewrite the expression.

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{(3 x - 1) + \sqrt{3 x - 1} \cdot 5 + 5 \sqrt{3 x - 1} + 5 \cdot 5 - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.3.1.2.5

Simplify.

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{3 x - 1 + \sqrt{3 x - 1} \cdot 5 + 5 \sqrt{3 x - 1} + 5 \cdot 5 - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.3.1.3

Move $5$ to the left of $\sqrt{3 x - 1}$ .

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{3 x - 1 + 5 \cdot \sqrt{3 x - 1} + 5 \sqrt{3 x - 1} + 5 \cdot 5 - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.3.1.4

Multiply $5$ by $5$ .

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{3 x - 1 + 5 \sqrt{3 x - 1} + 5 \sqrt{3 x - 1} + 25 - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.3.2

Add $- 1$ and $25$ .

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{3 x + 24 + 5 \sqrt{3 x - 1} + 5 \sqrt{3 x - 1} - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.3.3

Add $5 \sqrt{3 x - 1}$ and $5 \sqrt{3 x - 1}$ .

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{3 x + 24 + 10 \sqrt{3 x - 1} - 10 (\sqrt{3 x - 1} + 5) + 26}{3}$

Step 5.2.4.4

Apply the distributive property.

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{3 x + 24 + 10 \sqrt{3 x - 1} - 10 \sqrt{3 x - 1} - 10 \cdot 5 + 26}{3}$

Step 5.2.4.5

Multiply $- 10$ by $5$ .

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{3 x + 24 + 10 \sqrt{3 x - 1} - 10 \sqrt{3 x - 1} - 50 + 26}{3}$

Step 5.2.5

Simplify terms.

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Step 5.2.5.1

Combine the opposite terms in $3 x + 24 + 10 \sqrt{3 x - 1} - 10 \sqrt{3 x - 1} - 50 + 26$ .

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Step 5.2.5.1.1

Subtract $10 \sqrt{3 x - 1}$ from $10 \sqrt{3 x - 1}$ .

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{3 x + 24 + 0 - 50 + 26}{3}$

Step 5.2.5.1.2

Add $3 x + 24$ and $0$ .

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{3 x + 24 - 50 + 26}{3}$

Step 5.2.5.2

Subtract $50$ from $24$ .

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{3 x - 26 + 26}{3}$

Step 5.2.5.3

Combine the opposite terms in $3 x - 26 + 26$ .

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Step 5.2.5.3.1

Add $- 26$ and $26$ .

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{3 x + 0}{3}$

Step 5.2.5.3.2

Add $3 x$ and $0$ .

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{3 x}{3}$

Step 5.2.5.4

Cancel the common factor of $3$ .

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Step 5.2.5.4.1

Cancel the common factor.

$f^{- 1} (\sqrt{3 x - 1} + 5) = \frac{3 x}{3}$

Step 5.2.5.4.2

Divide $x$ by $1$ .

$f^{- 1} (\sqrt{3 x - 1} + 5) = x$

Step 5.3

Evaluate $f (f^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate $f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) = \sqrt{3 (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) - 1} + 5$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) = \sqrt{3 (\frac{x^{2}}{3}) + 3 (- \frac{10 x}{3}) + 3 (\frac{26}{3}) - 1} + 5$

Step 5.3.3.2

Simplify.

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Step 5.3.3.2.1

Cancel the common factor of $3$ .

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Step 5.3.3.2.1.1

Cancel the common factor.

$f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) = \sqrt{3 (\frac{x^{2}}{3}) + 3 (- \frac{10 x}{3}) + 3 (\frac{26}{3}) - 1} + 5$

Step 5.3.3.2.1.2

Rewrite the expression.

$f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) = \sqrt{x^{2} + 3 (- \frac{10 x}{3}) + 3 (\frac{26}{3}) - 1} + 5$

Step 5.3.3.2.2

Cancel the common factor of $3$ .

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Step 5.3.3.2.2.1

Move the leading negative in $- \frac{10 x}{3}$ into the numerator.

$f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) = \sqrt{x^{2} + 3 (\frac{- 10 x}{3}) + 3 (\frac{26}{3}) - 1} + 5$

Step 5.3.3.2.2.2

Cancel the common factor.

$f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) = \sqrt{x^{2} + 3 (\frac{- 10 x}{3}) + 3 (\frac{26}{3}) - 1} + 5$

Step 5.3.3.2.2.3

Rewrite the expression.

$f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) = \sqrt{x^{2} - 10 x + 3 (\frac{26}{3}) - 1} + 5$

Step 5.3.3.2.3

Cancel the common factor of $3$ .

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Step 5.3.3.2.3.1

Cancel the common factor.

$f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) = \sqrt{x^{2} - 10 x + 3 (\frac{26}{3}) - 1} + 5$

Step 5.3.3.2.3.2

Rewrite the expression.

$f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) = \sqrt{x^{2} - 10 x + 26 - 1} + 5$

Step 5.3.3.3

Subtract $1$ from $26$ .

$f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) = \sqrt{x^{2} - 10 x + 25} + 5$

Step 5.3.3.4

Factor using the perfect square rule.

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Step 5.3.3.4.1

Rewrite $25$ as $5^{2}$ .

$f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) = \sqrt{x^{2} - 10 x + 5^{2}} + 5$

Step 5.3.3.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 5.3.3.4.3

Rewrite the polynomial.

$f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) = \sqrt{x^{2} - 2 \cdot x \cdot 5 + 5^{2}} + 5$

Step 5.3.3.4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 5$ .

$f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) = \sqrt{{(x - 5)}^{2}} + 5$

Step 5.3.3.5

Pull terms out from under the radical, assuming positive real numbers.

$f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) = x - 5 + 5$

Step 5.3.4

Combine the opposite terms in $x - 5 + 5$ .

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Step 5.3.4.1

Add $- 5$ and $5$ .

$f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}$ is the inverse of $f (x) = \sqrt{3 x - 1} + 5$ .

$f^{- 1} (x) = \frac{x^{2}}{3} - \frac{10 x}{3} + \frac{26}{3}$