Precalculus Examples

$\frac{1}{3} (x - 5)$

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{1}{3} \cdot (y - 5) = x$ .

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

Step 2.2

Multiply both sides of the equation by $3$ .

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

Step 2.3

Simplify the left side.

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Step 2.3.1

Simplify $3 (\frac{1}{3} \cdot (y - 5))$ .

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Step 2.3.1.1

Apply the distributive property.

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

Step 2.3.1.2

Combine $\frac{1}{3}$ and $y$ .

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

Step 2.3.1.3

Combine $\frac{1}{3}$ and $- 5$ .

$3 (\frac{y}{3} + \frac{- 5}{3}) = 3 x$

Step 2.3.1.4

Move the negative in front of the fraction.

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

Step 2.3.1.5

Apply the distributive property.

$3 \frac{y}{3} + 3 (- \frac{5}{3}) = 3 x$

Step 2.3.1.6

Cancel the common factor of $3$ .

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Step 2.3.1.6.1

Cancel the common factor.

$3 \frac{y}{3} + 3 (- \frac{5}{3}) = 3 x$

Step 2.3.1.6.2

Rewrite the expression.

$y + 3 (- \frac{5}{3}) = 3 x$

Step 2.3.1.7

Cancel the common factor of $3$ .

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Step 2.3.1.7.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$y + 3 (\frac{- 5}{3}) = 3 x$

Step 2.3.1.7.2

Cancel the common factor.

$y + 3 (\frac{- 5}{3}) = 3 x$

Step 2.3.1.7.3

Rewrite the expression.

$y - 5 = 3 x$

Step 2.4

Add $5$ to both sides of the equation.

$y = 3 x + 5$

Step 3

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

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

Step 4

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

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

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

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Apply the distributive property.

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

Step 4.2.3.2

Combine $\frac{1}{3}$ and $x$ .

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

Step 4.2.3.3

Combine $\frac{1}{3}$ and $- 5$ .

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

Step 4.2.3.4

Move the negative in front of the fraction.

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

Step 4.2.3.5

Apply the distributive property.

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

Step 4.2.3.6

Cancel the common factor of $3$ .

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Step 4.2.3.6.1

Cancel the common factor.

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

Step 4.2.3.6.2

Rewrite the expression.

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

Step 4.2.3.7

Cancel the common factor of $3$ .

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Step 4.2.3.7.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

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

Step 4.2.3.7.2

Cancel the common factor.

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

Step 4.2.3.7.3

Rewrite the expression.

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

Step 4.2.4

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

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Step 4.2.4.1

Add $- 5$ and $5$ .

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

Step 4.2.4.2

Add $x$ and $0$ .

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

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

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

Step 4.3.3

Combine the opposite terms in $(3 x + 5) - 5$ .

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Step 4.3.3.1

Subtract $5$ from $5$ .

$f (3 x + 5) = \frac{1}{3} \cdot (3 x + 0)$

Step 4.3.3.2

Add $3 x$ and $0$ .

$f (3 x + 5) = \frac{1}{3} \cdot (3 x)$

Step 4.3.4

Cancel the common factor of $3$ .

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Step 4.3.4.1

Factor $3$ out of $3 x$ .

$f (3 x + 5) = \frac{1}{3} \cdot (3 (x))$

Step 4.3.4.2

Cancel the common factor.

$f (3 x + 5) = \frac{1}{3} \cdot (3 x)$

Step 4.3.4.3

Rewrite the expression.

$f (3 x + 5) = x$

Step 4.4

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

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