Precalculus Examples

$f (x) = \frac{5}{2} x + 3$

Step 1

Write $f (x) = \frac{5}{2} x + 3$ as an equation.

$y = \frac{5}{2} x + 3$

Step 2

Interchange the variables.

$x = \frac{5}{2} y + 3$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{5}{2} y + 3 = x$ .

$\frac{5}{2} y + 3 = x$

Step 3.2

Combine $\frac{5}{2}$ and $y$ .

$\frac{5 y}{2} + 3 = x$

Step 3.3

Subtract $3$ from both sides of the equation.

$\frac{5 y}{2} = x - 3$

Step 3.4

Multiply both sides of the equation by $\frac{2}{5}$ .

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

Step 3.5

Simplify both sides of the equation.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Simplify $\frac{2}{5} \cdot \frac{5 y}{2}$ .

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Step 3.5.1.1.1

Cancel the common factor of $2$ .

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Step 3.5.1.1.1.1

Cancel the common factor.

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

Step 3.5.1.1.1.2

Rewrite the expression.

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

Step 3.5.1.1.2

Cancel the common factor of $5$ .

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Step 3.5.1.1.2.1

Factor $5$ out of $5 y$ .

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

Step 3.5.1.1.2.2

Cancel the common factor.

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

Step 3.5.1.1.2.3

Rewrite the expression.

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

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $\frac{2}{5} (x - 3)$ .

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Step 3.5.2.1.1

Apply the distributive property.

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

Step 3.5.2.1.2

Combine $\frac{2}{5}$ and $x$ .

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

Step 3.5.2.1.3

Multiply $\frac{2}{5} \cdot - 3$ .

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Step 3.5.2.1.3.1

Combine $\frac{2}{5}$ and $- 3$ .

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

Step 3.5.2.1.3.2

Multiply $2$ by $- 3$ .

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

Step 3.5.2.1.4

Move the negative in front of the fraction.

$y = \frac{2 x}{5} - \frac{6}{5}$

Step 4

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

$f^{- 1} (x) = \frac{2 x}{5} - \frac{6}{5}$

Step 5

Verify if $f^{- 1} (x) = \frac{2 x}{5} - \frac{6}{5}$ is the inverse of $f (x) = \frac{5}{2} x + 3$ .

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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 $f^{- 1} (\frac{5}{2} x + 3)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{5}{2} x + 3) = \frac{2 (\frac{5}{2} x + 3)}{5} - \frac{6}{5}$

Step 5.2.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{5}{2} x + 3) = \frac{2 (\frac{5}{2} x + 3) - 6}{5}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Combine $\frac{5}{2}$ and $x$ .

$f^{- 1} (\frac{5}{2} x + 3) = \frac{2 (\frac{5 x}{2} + 3) - 6}{5}$

Step 5.2.4.2

Apply the distributive property.

$f^{- 1} (\frac{5}{2} x + 3) = \frac{2 (\frac{5 x}{2}) + 2 \cdot 3 - 6}{5}$

Step 5.2.4.3

Cancel the common factor of $2$ .

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Step 5.2.4.3.1

Cancel the common factor.

$f^{- 1} (\frac{5}{2} x + 3) = \frac{2 (\frac{5 x}{2}) + 2 \cdot 3 - 6}{5}$

Step 5.2.4.3.2

Rewrite the expression.

$f^{- 1} (\frac{5}{2} x + 3) = \frac{5 x + 2 \cdot 3 - 6}{5}$

Step 5.2.4.4

Multiply $2$ by $3$ .

$f^{- 1} (\frac{5}{2} x + 3) = \frac{5 x + 6 - 6}{5}$

Step 5.2.5

Simplify terms.

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Step 5.2.5.1

Combine the opposite terms in $5 x + 6 - 6$ .

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Step 5.2.5.1.1

Subtract $6$ from $6$ .

$f^{- 1} (\frac{5}{2} x + 3) = \frac{5 x + 0}{5}$

Step 5.2.5.1.2

Add $5 x$ and $0$ .

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

Step 5.2.5.2

Cancel the common factor of $5$ .

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Step 5.2.5.2.1

Cancel the common factor.

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

Step 5.2.5.2.2

Divide $x$ by $1$ .

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

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 $f (\frac{2 x}{5} - \frac{6}{5})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{2 x}{5} - \frac{6}{5}) = \frac{5}{2} \cdot (\frac{2 x}{5} - \frac{6}{5}) + 3$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (\frac{2 x}{5} - \frac{6}{5}) = \frac{5}{2} \cdot \frac{2 x}{5} + \frac{5}{2} \cdot (- \frac{6}{5}) + 3$

Step 5.3.3.2

Cancel the common factor of $5$ .

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Step 5.3.3.2.1

Cancel the common factor.

$f (\frac{2 x}{5} - \frac{6}{5}) = \frac{5}{2} \cdot \frac{2 x}{5} + \frac{5}{2} \cdot (- \frac{6}{5}) + 3$

Step 5.3.3.2.2

Rewrite the expression.

$f (\frac{2 x}{5} - \frac{6}{5}) = \frac{1}{2} \cdot (2 x) + \frac{5}{2} \cdot (- \frac{6}{5}) + 3$

Step 5.3.3.3

Cancel the common factor of $2$ .

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Step 5.3.3.3.1

Factor $2$ out of $2 x$ .

$f (\frac{2 x}{5} - \frac{6}{5}) = \frac{1}{2} \cdot (2 (x)) + \frac{5}{2} \cdot (- \frac{6}{5}) + 3$

Step 5.3.3.3.2

Cancel the common factor.

$f (\frac{2 x}{5} - \frac{6}{5}) = \frac{1}{2} \cdot (2 x) + \frac{5}{2} \cdot (- \frac{6}{5}) + 3$

Step 5.3.3.3.3

Rewrite the expression.

$f (\frac{2 x}{5} - \frac{6}{5}) = x + \frac{5}{2} \cdot (- \frac{6}{5}) + 3$

Step 5.3.3.4

Cancel the common factor of $5$ .

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Step 5.3.3.4.1

Move the leading negative in $- \frac{6}{5}$ into the numerator.

$f (\frac{2 x}{5} - \frac{6}{5}) = x + \frac{5}{2} \cdot \frac{- 6}{5} + 3$

Step 5.3.3.4.2

Cancel the common factor.

$f (\frac{2 x}{5} - \frac{6}{5}) = x + \frac{5}{2} \cdot \frac{- 6}{5} + 3$

Step 5.3.3.4.3

Rewrite the expression.

$f (\frac{2 x}{5} - \frac{6}{5}) = x + \frac{1}{2} \cdot - 6 + 3$

Step 5.3.3.5

Cancel the common factor of $2$ .

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Step 5.3.3.5.1

Factor $2$ out of $- 6$ .

$f (\frac{2 x}{5} - \frac{6}{5}) = x + \frac{1}{2} \cdot (2 (- 3)) + 3$

Step 5.3.3.5.2

Cancel the common factor.

$f (\frac{2 x}{5} - \frac{6}{5}) = x + \frac{1}{2} \cdot (2 \cdot - 3) + 3$

Step 5.3.3.5.3

Rewrite the expression.

$f (\frac{2 x}{5} - \frac{6}{5}) = x - 3 + 3$

Step 5.3.4

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

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Step 5.3.4.1

Add $- 3$ and $3$ .

$f (\frac{2 x}{5} - \frac{6}{5}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{2 x}{5} - \frac{6}{5}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{2 x}{5} - \frac{6}{5}$ is the inverse of $f (x) = \frac{5}{2} x + 3$ .

$f^{- 1} (x) = \frac{2 x}{5} - \frac{6}{5}$