Calculus Examples

$f (x) = 6 x + 15$

Step 1

Write $f (x) = 6 x + 15$ as an equation.

$y = 6 x + 15$

Step 2

Interchange the variables.

$x = 6 y + 15$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $6 y + 15 = x$ .

$6 y + 15 = x$

Step 3.2

Subtract $15$ from both sides of the equation.

$6 y = x - 15$

Step 3.3

Divide each term in $6 y = x - 15$ by $6$ and simplify.

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Step 3.3.1

Divide each term in $6 y = x - 15$ by $6$ .

$\frac{6 y}{6} = \frac{x}{6} + \frac{- 15}{6}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $6$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{x}{6} + \frac{- 15}{6}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{6} + \frac{- 15}{6}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Cancel the common factor of $- 15$ and $6$ .

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Step 3.3.3.1.1.1

Factor $3$ out of $- 15$ .

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

Step 3.3.3.1.1.2

Cancel the common factors.

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Step 3.3.3.1.1.2.1

Factor $3$ out of $6$ .

$y = \frac{x}{6} + \frac{3 \cdot - 5}{3 \cdot 2}$

Step 3.3.3.1.1.2.2

Cancel the common factor.

$y = \frac{x}{6} + \frac{3 \cdot - 5}{3 \cdot 2}$

Step 3.3.3.1.1.2.3

Rewrite the expression.

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

Step 3.3.3.1.2

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{x}{6} - \frac{5}{2}$ is the inverse of $f (x) = 6 x + 15$ .

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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 $f^{- 1} (6 x + 15)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (6 x + 15) = \frac{6 x + 15}{6} - \frac{5}{2}$

Step 5.2.3

Cancel the common factor of $6 x + 15$ and $6$ .

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Step 5.2.3.1

Factor $3$ out of $6 x$ .

$f^{- 1} (6 x + 15) = \frac{3 (2 x) + 15}{6} - \frac{5}{2}$

Step 5.2.3.2

Factor $3$ out of $15$ .

$f^{- 1} (6 x + 15) = \frac{3 (2 x) + 3 (5)}{6} - \frac{5}{2}$

Step 5.2.3.3

Factor $3$ out of $3 (2 x) + 3 (5)$ .

$f^{- 1} (6 x + 15) = \frac{3 (2 x + 5)}{6} - \frac{5}{2}$

Step 5.2.3.4

Cancel the common factors.

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Step 5.2.3.4.1

Factor $3$ out of $6$ .

$f^{- 1} (6 x + 15) = \frac{3 (2 x + 5)}{3 (2)} - \frac{5}{2}$

Step 5.2.3.4.2

Cancel the common factor.

$f^{- 1} (6 x + 15) = \frac{3 (2 x + 5)}{3 \cdot 2} - \frac{5}{2}$

Step 5.2.3.4.3

Rewrite the expression.

$f^{- 1} (6 x + 15) = \frac{2 x + 5}{2} - \frac{5}{2}$

Step 5.2.4

Combine the numerators over the common denominator.

$f^{- 1} (6 x + 15) = \frac{2 x + 5 - 5}{2}$

Step 5.2.5

Combine the opposite terms in $2 x + 5 - 5$ .

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Step 5.2.5.1

Subtract $5$ from $5$ .

$f^{- 1} (6 x + 15) = \frac{2 x + 0}{2}$

Step 5.2.5.2

Add $2 x$ and $0$ .

$f^{- 1} (6 x + 15) = \frac{2 x}{2}$

Step 5.2.6

Cancel the common factor of $2$ .

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Step 5.2.6.1

Cancel the common factor.

$f^{- 1} (6 x + 15) = \frac{2 x}{2}$

Step 5.2.6.2

Divide $x$ by $1$ .

$f^{- 1} (6 x + 15) = x$

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

$f (\frac{x}{6} - \frac{5}{2}) = 6 (\frac{x}{6} - \frac{5}{2}) + 15$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (\frac{x}{6} - \frac{5}{2}) = 6 (\frac{x}{6}) + 6 (- \frac{5}{2}) + 15$

Step 5.3.3.2

Cancel the common factor of $6$ .

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Step 5.3.3.2.1

Cancel the common factor.

$f (\frac{x}{6} - \frac{5}{2}) = 6 (\frac{x}{6}) + 6 (- \frac{5}{2}) + 15$

Step 5.3.3.2.2

Rewrite the expression.

$f (\frac{x}{6} - \frac{5}{2}) = x + 6 (- \frac{5}{2}) + 15$

Step 5.3.3.3

Cancel the common factor of $2$ .

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Step 5.3.3.3.1

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$f (\frac{x}{6} - \frac{5}{2}) = x + 6 (\frac{- 5}{2}) + 15$

Step 5.3.3.3.2

Factor $2$ out of $6$ .

$f (\frac{x}{6} - \frac{5}{2}) = x + 2 (3) (\frac{- 5}{2}) + 15$

Step 5.3.3.3.3

Cancel the common factor.

$f (\frac{x}{6} - \frac{5}{2}) = x + 2 \cdot (3 (\frac{- 5}{2})) + 15$

Step 5.3.3.3.4

Rewrite the expression.

$f (\frac{x}{6} - \frac{5}{2}) = x + 3 \cdot - 5 + 15$

Step 5.3.3.4

Multiply $3$ by $- 5$ .

$f (\frac{x}{6} - \frac{5}{2}) = x - 15 + 15$

Step 5.3.4

Combine the opposite terms in $x - 15 + 15$ .

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Step 5.3.4.1

Add $- 15$ and $15$ .

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

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{6} - \frac{5}{2}$ is the inverse of $f (x) = 6 x + 15$ .

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