Precalculus Examples

$y = \frac{1}{2} x + 3$

Step 1

Interchange the variables.

$x = \frac{1}{2} y + 3$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{1}{2} y + 3 = x$ .

$\frac{1}{2} y + 3 = x$

Step 2.2

Combine $\frac{1}{2}$ and $y$ .

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

Step 2.3

Subtract $3$ from both sides of the equation.

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

Step 2.4

Multiply both sides of the equation by $2$ .

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

Step 2.5

Simplify both sides of the equation.

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Step 2.5.1

Simplify the left side.

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Step 2.5.1.1

Cancel the common factor of $2$ .

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Step 2.5.1.1.1

Cancel the common factor.

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

Step 2.5.1.1.2

Rewrite the expression.

$y = 2 (x - 3)$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify $2 (x - 3)$ .

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Step 2.5.2.1.1

Apply the distributive property.

$y = 2 x + 2 \cdot - 3$

Step 2.5.2.1.2

Multiply $2$ by $- 3$ .

$y = 2 x - 6$

Step 3

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

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

Step 4

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

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

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

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Combine $\frac{1}{2}$ and $x$ .

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

Step 4.2.3.2

Apply the distributive property.

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

Step 4.2.3.3

Cancel the common factor of $2$ .

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Step 4.2.3.3.1

Cancel the common factor.

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

Step 4.2.3.3.2

Rewrite the expression.

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

Step 4.2.3.4

Multiply $2$ by $3$ .

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

Step 4.2.4

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

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Step 4.2.4.1

Subtract $6$ from $6$ .

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

Step 4.2.4.2

Add $x$ and $0$ .

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

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

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

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

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

Step 4.3.3.2

Cancel the common factor of $2$ .

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Step 4.3.3.2.1

Factor $2$ out of $2 x$ .

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

Step 4.3.3.2.2

Cancel the common factor.

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

Step 4.3.3.2.3

Rewrite the expression.

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

Step 4.3.3.3

Cancel the common factor of $2$ .

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Step 4.3.3.3.1

Factor $2$ out of $- 6$ .

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

Step 4.3.3.3.2

Cancel the common factor.

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

Step 4.3.3.3.3

Rewrite the expression.

$f (2 x - 6) = x - 3 + 3$

Step 4.3.4

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

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Step 4.3.4.1

Add $- 3$ and $3$ .

$f (2 x - 6) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (2 x - 6) = x$

Step 4.4

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

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