Trigonometry Examples

$f (x) = \frac{x - 24}{4}$

Step 1

Write $f (x) = \frac{x - 24}{4}$ as an equation.

$y = \frac{x - 24}{4}$

Step 2

Interchange the variables.

$x = \frac{y - 24}{4}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{y - 24}{4} = x$ .

$\frac{y - 24}{4} = x$

Step 3.2

Multiply both sides of the equation by $4$ .

$4 \frac{y - 24}{4} = 4 x$

Step 3.3

Simplify the left side.

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Step 3.3.1

Cancel the common factor of $4$ .

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Step 3.3.1.1

Cancel the common factor.

$4 \frac{y - 24}{4} = 4 x$

Step 3.3.1.2

Rewrite the expression.

$y - 24 = 4 x$

Step 3.4

Add $24$ to both sides of the equation.

$y = 4 x + 24$

Step 4

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

$f^{- 1} (x) = 4 x + 24$

Step 5

Verify if $f^{- 1} (x) = 4 x + 24$ is the inverse of $f (x) = \frac{x - 24}{4}$ .

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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 $f^{- 1} (\frac{x - 24}{4})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x - 24}{4}) = 4 (\frac{x - 24}{4}) + 24$

Step 5.2.3

Cancel the common factor of $4$ .

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Step 5.2.3.1

Cancel the common factor.

$f^{- 1} (\frac{x - 24}{4}) = 4 (\frac{x - 24}{4}) + 24$

Step 5.2.3.2

Rewrite the expression.

$f^{- 1} (\frac{x - 24}{4}) = x - 24 + 24$

Step 5.2.4

Combine the opposite terms in $x - 24 + 24$ .

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Step 5.2.4.1

Add $- 24$ and $24$ .

$f^{- 1} (\frac{x - 24}{4}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{x - 24}{4}) = x$

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 $f (4 x + 24)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (4 x + 24) = \frac{(4 x + 24) - 24}{4}$

Step 5.3.3

Cancel the common factor of $(4 x + 24) - 24$ and $4$ .

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Step 5.3.3.1

Factor $4$ out of $4 x$ .

$f (4 x + 24) = \frac{4 (x) + 24 - 24}{4}$

Step 5.3.3.2

Factor $4$ out of $24$ .

$f (4 x + 24) = \frac{4 (x) + 4 \cdot 6 - 24}{4}$

Step 5.3.3.3

Factor $4$ out of $4 (x) + 4 (6)$ .

$f (4 x + 24) = \frac{4 (x + 6) - 24}{4}$

Step 5.3.3.4

Factor $4$ out of $- 24$ .

$f (4 x + 24) = \frac{4 (x + 6) + 4 \cdot - 6}{4}$

Step 5.3.3.5

Factor $4$ out of $4 (x + 6) + 4 (- 6)$ .

$f (4 x + 24) = \frac{4 (x + 6 - 6)}{4}$

Step 5.3.3.6

Cancel the common factors.

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Step 5.3.3.6.1

Factor $4$ out of $4$ .

$f (4 x + 24) = \frac{4 (x + 6 - 6)}{4 (1)}$

Step 5.3.3.6.2

Cancel the common factor.

$f (4 x + 24) = \frac{4 (x + 6 - 6)}{4 \cdot 1}$

Step 5.3.3.6.3

Rewrite the expression.

$f (4 x + 24) = \frac{x + 6 - 6}{1}$

Step 5.3.3.6.4

Divide $x + 6 - 6$ by $1$ .

$f (4 x + 24) = x + 6 - 6$

Step 5.3.4

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

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Step 5.3.4.1

Subtract $6$ from $6$ .

$f (4 x + 24) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (4 x + 24) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 4 x + 24$ is the inverse of $f (x) = \frac{x - 24}{4}$ .

$f^{- 1} (x) = 4 x + 24$