Trigonometry Examples

$f (x) = 25 + 0.08 x$

Step 1

Write $f (x) = 25 + 0.08 x$ as an equation.

$y = 25 + 0.08 x$

Step 2

Interchange the variables.

$x = 25 + 0.08 y$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $25 + 0.08 y = x$ .

$25 + 0.08 y = x$

Step 3.2

Subtract $25$ from both sides of the equation.

$0.08 y = x - 25$

Step 3.3

Divide each term in $0.08 y = x - 25$ by $0.08$ and simplify.

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Step 3.3.1

Divide each term in $0.08 y = x - 25$ by $0.08$ .

$\frac{0.08 y}{0.08} = \frac{x}{0.08} + \frac{- 25}{0.08}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $0.08$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{0.08 y}{0.08} = \frac{x}{0.08} + \frac{- 25}{0.08}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{0.08} + \frac{- 25}{0.08}$

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

Multiply by $1$ .

$y = \frac{1 x}{0.08} + \frac{- 25}{0.08}$

Step 3.3.3.1.2

Factor $0.08$ out of $0.08$ .

$y = \frac{1 x}{0.08 (1)} + \frac{- 25}{0.08}$

Step 3.3.3.1.3

Separate fractions.

$y = \frac{1}{0.08} \cdot \frac{x}{1} + \frac{- 25}{0.08}$

Step 3.3.3.1.4

Divide $1$ by $0.08$ .

$y = 12.5 \frac{x}{1} + \frac{- 25}{0.08}$

Step 3.3.3.1.5

Divide $x$ by $1$ .

$y = 12.5 x + \frac{- 25}{0.08}$

Step 3.3.3.1.6

Divide $- 25$ by $0.08$ .

$y = 12.5 x - 312.5$

Step 4

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

$f^{- 1} (x) = 12.5 x - 312.5$

Step 5

Verify if $f^{- 1} (x) = 12.5 x - 312.5$ is the inverse of $f (x) = 25 + 0.08 x$ .

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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 $f^{- 1} (25 + 0.08 x)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (25 + 0.08 x) = 12.5 (25 + 0.08 x) - 312.5$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Apply the distributive property.

$f^{- 1} (25 + 0.08 x) = 12.5 \cdot 25 + 12.5 (0.08 x) - 312.5$

Step 5.2.3.2

Multiply $12.5$ by $25$ .

$f^{- 1} (25 + 0.08 x) = 312.5 + 12.5 (0.08 x) - 312.5$

Step 5.2.3.3

Multiply $12.5 (0.08 x)$ .

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Step 5.2.3.3.1

Multiply $0.08$ by $12.5$ .

$f^{- 1} (25 + 0.08 x) = 312.5 + 1 x - 312.5$

Step 5.2.3.3.2

Multiply $x$ by $1$ .

$f^{- 1} (25 + 0.08 x) = 312.5 + x - 312.5$

Step 5.2.4

Combine the opposite terms in $312.5 + x - 312.5$ .

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Step 5.2.4.1

Subtract $312.5$ from $312.5$ .

$f^{- 1} (25 + 0.08 x) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (25 + 0.08 x) = x$

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 $f (12.5 x - 312.5)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (12.5 x - 312.5) = 25 + 0.08 (12.5 x - 312.5)$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (12.5 x - 312.5) = 25 + 0.08 (12.5 x) + 0.08 \cdot - 312.5$

Step 5.3.3.2

Multiply $0.08 (12.5 x)$ .

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Step 5.3.3.2.1

Multiply $12.5$ by $0.08$ .

$f (12.5 x - 312.5) = 25 + 1 x + 0.08 \cdot - 312.5$

Step 5.3.3.2.2

Multiply $x$ by $1$ .

$f (12.5 x - 312.5) = 25 + x + 0.08 \cdot - 312.5$

Step 5.3.3.3

Multiply $0.08$ by $- 312.5$ .

$f (12.5 x - 312.5) = 25 + x - 25$

Step 5.3.4

Simplify by subtracting numbers.

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Step 5.3.4.1

Subtract $25$ from $25$ .

$f (12.5 x - 312.5) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (12.5 x - 312.5) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 12.5 x - 312.5$ is the inverse of $f (x) = 25 + 0.08 x$ .

$f^{- 1} (x) = 12.5 x - 312.5$