Calculus Examples

$f (x) = 18 - 6 x$

Step 1

Write $f (x) = 18 - 6 x$ as an equation.

$y = 18 - 6 x$

Step 2

Interchange the variables.

$x = 18 - 6 y$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $18 - 6 y = x$ .

$18 - 6 y = x$

Step 3.2

Subtract $18$ from both sides of the equation.

$- 6 y = x - 18$

Step 3.3

Divide each term in $- 6 y = x - 18$ by $- 6$ and simplify.

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Step 3.3.1

Divide each term in $- 6 y = x - 18$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{x}{- 6} + \frac{- 18}{- 6}$

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.

$\frac{- 6 y}{- 6} = \frac{x}{- 6} + \frac{- 18}{- 6}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{- 6} + \frac{- 18}{- 6}$

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

Move the negative in front of the fraction.

$y = - \frac{x}{6} + \frac{- 18}{- 6}$

Step 3.3.3.1.2

Divide $- 18$ by $- 6$ .

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

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = - \frac{x}{6} + 3$ is the inverse of $f (x) = 18 - 6 x$ .

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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 $f^{- 1} (18 - 6 x)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (18 - 6 x) = - \frac{18 - 6 x}{6} + 3$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Cancel the common factor of $18 - 6 x$ and $6$ .

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Step 5.2.3.1.1

Factor $6$ out of $18$ .

$f^{- 1} (18 - 6 x) = - \frac{6 \cdot 3 - 6 x}{6} + 3$

Step 5.2.3.1.2

Factor $6$ out of $- 6 x$ .

$f^{- 1} (18 - 6 x) = - \frac{6 \cdot 3 + 6 (- x)}{6} + 3$

Step 5.2.3.1.3

Factor $6$ out of $6 (3) + 6 (- x)$ .

$f^{- 1} (18 - 6 x) = - \frac{6 (3 - x)}{6} + 3$

Step 5.2.3.1.4

Cancel the common factors.

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Step 5.2.3.1.4.1

Factor $6$ out of $6$ .

$f^{- 1} (18 - 6 x) = - \frac{6 (3 - x)}{6 (1)} + 3$

Step 5.2.3.1.4.2

Cancel the common factor.

$f^{- 1} (18 - 6 x) = - \frac{6 (3 - x)}{6 \cdot 1} + 3$

Step 5.2.3.1.4.3

Rewrite the expression.

$f^{- 1} (18 - 6 x) = - \frac{3 - x}{1} + 3$

Step 5.2.3.1.4.4

Divide $3 - x$ by $1$ .

$f^{- 1} (18 - 6 x) = - (3 - x) + 3$

Step 5.2.3.2

Apply the distributive property.

$f^{- 1} (18 - 6 x) = - 1 \cdot 3 + x + 3$

Step 5.2.3.3

Multiply $- 1$ by $3$ .

$f^{- 1} (18 - 6 x) = - 3 + x + 3$

Step 5.2.3.4

Multiply $- - x$ .

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Step 5.2.3.4.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (18 - 6 x) = - 3 + 1 x + 3$

Step 5.2.3.4.2

Multiply $x$ by $1$ .

$f^{- 1} (18 - 6 x) = - 3 + x + 3$

Step 5.2.4

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

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Step 5.2.4.1

Add $- 3$ and $3$ .

$f^{- 1} (18 - 6 x) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (18 - 6 x) = x$

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 $f (- \frac{x}{6} + 3)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{x}{6} + 3) = 18 - 6 (- \frac{x}{6} + 3)$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (- \frac{x}{6} + 3) = 18 - 6 (- \frac{x}{6}) - 6 \cdot 3$

Step 5.3.3.2

Cancel the common factor of $6$ .

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Step 5.3.3.2.1

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

$f (- \frac{x}{6} + 3) = 18 - 6 \frac{- x}{6} - 6 \cdot 3$

Step 5.3.3.2.2

Factor $6$ out of $- 6$ .

$f (- \frac{x}{6} + 3) = 18 + 6 (- 1) (\frac{- x}{6}) - 6 \cdot 3$

Step 5.3.3.2.3

Cancel the common factor.

$f (- \frac{x}{6} + 3) = 18 + 6 \cdot (- 1 \frac{- x}{6}) - 6 \cdot 3$

Step 5.3.3.2.4

Rewrite the expression.

$f (- \frac{x}{6} + 3) = 18 - 1 (- x) - 6 \cdot 3$

Step 5.3.3.3

Multiply $- 1$ by $- 1$ .

$f (- \frac{x}{6} + 3) = 18 + 1 x - 6 \cdot 3$

Step 5.3.3.4

Multiply $x$ by $1$ .

$f (- \frac{x}{6} + 3) = 18 + x - 6 \cdot 3$

Step 5.3.3.5

Multiply $- 6$ by $3$ .

$f (- \frac{x}{6} + 3) = 18 + x - 18$

Step 5.3.4

Combine the opposite terms in $18 + x - 18$ .

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Step 5.3.4.1

Subtract $18$ from $18$ .

$f (- \frac{x}{6} + 3) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{x}{6} + 3$ is the inverse of $f (x) = 18 - 6 x$ .

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