Precalculus Examples

$y = 2 x - 8$

Step 1

Interchange the variables.

$x = 2 y - 8$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $2 y - 8 = x$ .

$2 y - 8 = x$

Step 2.2

Add $8$ to both sides of the equation.

$2 y = x + 8$

Step 2.3

Divide each term in $2 y = x + 8$ by $2$ and simplify.

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Step 2.3.1

Divide each term in $2 y = x + 8$ by $2$ .

$\frac{2 y}{2} = \frac{x}{2} + \frac{8}{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{x}{2} + \frac{8}{2}$

Step 2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{2} + \frac{8}{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $8$ by $2$ .

$y = \frac{x}{2} + 4$

Step 3

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

$f^{- 1} (x) = \frac{x}{2} + 4$

Step 4

Verify if $f^{- 1} (x) = \frac{x}{2} + 4$ is the inverse of $f (x) = 2 x - 8$ .

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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 $f^{- 1} (2 x - 8)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (2 x - 8) = \frac{2 x - 8}{2} + 4$

Step 4.2.3

Cancel the common factor of $2 x - 8$ and $2$ .

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Step 4.2.3.1

Factor $2$ out of $2 x$ .

$f^{- 1} (2 x - 8) = \frac{2 (x) - 8}{2} + 4$

Step 4.2.3.2

Factor $2$ out of $- 8$ .

$f^{- 1} (2 x - 8) = \frac{2 (x) + 2 \cdot - 4}{2} + 4$

Step 4.2.3.3

Factor $2$ out of $2 (x) + 2 (- 4)$ .

$f^{- 1} (2 x - 8) = \frac{2 (x - 4)}{2} + 4$

Step 4.2.3.4

Cancel the common factors.

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Step 4.2.3.4.1

Factor $2$ out of $2$ .

$f^{- 1} (2 x - 8) = \frac{2 (x - 4)}{2 (1)} + 4$

Step 4.2.3.4.2

Cancel the common factor.

$f^{- 1} (2 x - 8) = \frac{2 (x - 4)}{2 \cdot 1} + 4$

Step 4.2.3.4.3

Rewrite the expression.

$f^{- 1} (2 x - 8) = \frac{x - 4}{1} + 4$

Step 4.2.3.4.4

Divide $x - 4$ by $1$ .

$f^{- 1} (2 x - 8) = x - 4 + 4$

Step 4.2.4

Combine the opposite terms in $x - 4 + 4$ .

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Step 4.2.4.1

Add $- 4$ and $4$ .

$f^{- 1} (2 x - 8) = x + 0$

Step 4.2.4.2

Add $x$ and $0$ .

$f^{- 1} (2 x - 8) = x$

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 $f (\frac{x}{2} + 4)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{2} + 4) = 2 (\frac{x}{2} + 4) - 8$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

$f (\frac{x}{2} + 4) = 2 (\frac{x}{2}) + 2 \cdot 4 - 8$

Step 4.3.3.2

Cancel the common factor of $2$ .

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Step 4.3.3.2.1

Cancel the common factor.

$f (\frac{x}{2} + 4) = 2 (\frac{x}{2}) + 2 \cdot 4 - 8$

Step 4.3.3.2.2

Rewrite the expression.

$f (\frac{x}{2} + 4) = x + 2 \cdot 4 - 8$

Step 4.3.3.3

Multiply $2$ by $4$ .

$f (\frac{x}{2} + 4) = x + 8 - 8$

Step 4.3.4

Combine the opposite terms in $x + 8 - 8$ .

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Step 4.3.4.1

Subtract $8$ from $8$ .

$f (\frac{x}{2} + 4) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (\frac{x}{2} + 4) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{2} + 4$ is the inverse of $f (x) = 2 x - 8$ .

$f^{- 1} (x) = \frac{x}{2} + 4$