Trigonometry Examples

$f (x) = \frac{5}{2} x - 5$

Step 1

Write $f (x) = \frac{5}{2} x - 5$ as an equation.

$y = \frac{5}{2} x - 5$

Step 2

Interchange the variables.

$x = \frac{5}{2} y - 5$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{5}{2} y - 5 = x$ .

$\frac{5}{2} y - 5 = x$

Step 3.2

Combine $\frac{5}{2}$ and $y$ .

$\frac{5 y}{2} - 5 = x$

Step 3.3

Add $5$ to both sides of the equation.

$\frac{5 y}{2} = x + 5$

Step 3.4

Multiply both sides of the equation by $\frac{2}{5}$ .

$\frac{2}{5} \cdot \frac{5 y}{2} = \frac{2}{5} (x + 5)$

Step 3.5

Simplify both sides of the equation.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Simplify $\frac{2}{5} \cdot \frac{5 y}{2}$ .

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Step 3.5.1.1.1

Cancel the common factor of $2$ .

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Step 3.5.1.1.1.1

Cancel the common factor.

$\frac{2}{5} \cdot \frac{5 y}{2} = \frac{2}{5} (x + 5)$

Step 3.5.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 y) = \frac{2}{5} (x + 5)$

Step 3.5.1.1.2

Cancel the common factor of $5$ .

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Step 3.5.1.1.2.1

Factor $5$ out of $5 y$ .

$\frac{1}{5} (5 (y)) = \frac{2}{5} (x + 5)$

Step 3.5.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 y) = \frac{2}{5} (x + 5)$

Step 3.5.1.1.2.3

Rewrite the expression.

$y = \frac{2}{5} (x + 5)$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $\frac{2}{5} (x + 5)$ .

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Step 3.5.2.1.1

Apply the distributive property.

$y = \frac{2}{5} x + \frac{2}{5} \cdot 5$

Step 3.5.2.1.2

Combine $\frac{2}{5}$ and $x$ .

$y = \frac{2 x}{5} + \frac{2}{5} \cdot 5$

Step 3.5.2.1.3

Cancel the common factor of $5$ .

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Step 3.5.2.1.3.1

Cancel the common factor.

$y = \frac{2 x}{5} + \frac{2}{5} \cdot 5$

Step 3.5.2.1.3.2

Rewrite the expression.

$y = \frac{2 x}{5} + 2$

Step 4

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

$f^{- 1} (x) = \frac{2 x}{5} + 2$

Step 5

Verify if $f^{- 1} (x) = \frac{2 x}{5} + 2$ is the inverse of $f (x) = \frac{5}{2} x - 5$ .

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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 $f^{- 1} (\frac{5}{2} x - 5)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{5}{2} x - 5) = \frac{2 (\frac{5}{2} x - 5)}{5} + 2$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

Combine $\frac{5}{2}$ and $x$ .

$f^{- 1} (\frac{5}{2} x - 5) = \frac{2 (\frac{5 x}{2} - 5)}{5} + 2$

Step 5.2.3.1.2

Factor $5$ out of $\frac{5 x}{2} - 5$ .

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Step 5.2.3.1.2.1

Factor $5$ out of $\frac{5 x}{2}$ .

$f^{- 1} (\frac{5}{2} x - 5) = \frac{2 (5 (\frac{x}{2}) - 5)}{5} + 2$

Step 5.2.3.1.2.2

Factor $5$ out of $- 5$ .

$f^{- 1} (\frac{5}{2} x - 5) = \frac{2 (5 (\frac{x}{2}) + 5 (- 1))}{5} + 2$

Step 5.2.3.1.2.3

Factor $5$ out of $5 \frac{x}{2} + 5 (- 1)$ .

$f^{- 1} (\frac{5}{2} x - 5) = \frac{2 (5 (\frac{x}{2} - 1))}{5} + 2$

$f^{- 1} (\frac{5}{2} x - 5) = \frac{2 \cdot (5 (\frac{x}{2} - 1))}{5} + 2$

Step 5.2.3.1.3

Multiply $2$ by $5$ .

$f^{- 1} (\frac{5}{2} x - 5) = \frac{10 (\frac{x}{2} - 1)}{5} + 2$

Step 5.2.3.1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f^{- 1} (\frac{5}{2} x - 5) = \frac{10 (\frac{x}{2} - 1 \cdot \frac{2}{2})}{5} + 2$

Step 5.2.3.1.5

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

$f^{- 1} (\frac{5}{2} x - 5) = \frac{10 (\frac{x}{2} + \frac{- 1 \cdot 2}{2})}{5} + 2$

Step 5.2.3.1.6

Combine the numerators over the common denominator.

$f^{- 1} (\frac{5}{2} x - 5) = \frac{10 (\frac{x - 1 \cdot 2}{2})}{5} + 2$

Step 5.2.3.1.7

Multiply $- 1$ by $2$ .

$f^{- 1} (\frac{5}{2} x - 5) = \frac{10 (\frac{x - 2}{2})}{5} + 2$

Step 5.2.3.2

Combine $10$ and $\frac{x - 2}{2}$ .

$f^{- 1} (\frac{5}{2} x - 5) = \frac{\frac{10 (x - 2)}{2}}{5} + 2$

Step 5.2.3.3

Reduce the expression by cancelling the common factors.

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Step 5.2.3.3.1

Reduce the expression $\frac{10 (x - 2)}{2}$ by cancelling the common factors.

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Step 5.2.3.3.1.1

Factor $2$ out of $10 (x - 2)$ .

$f^{- 1} (\frac{5}{2} x - 5) = \frac{\frac{2 (5 (x - 2))}{2}}{5} + 2$

Step 5.2.3.3.1.2

Factor $2$ out of $2$ .

$f^{- 1} (\frac{5}{2} x - 5) = \frac{\frac{2 (5 (x - 2))}{2 (1)}}{5} + 2$

Step 5.2.3.3.1.3

Cancel the common factor.

$f^{- 1} (\frac{5}{2} x - 5) = \frac{\frac{2 (5 (x - 2))}{2 \cdot 1}}{5} + 2$

Step 5.2.3.3.1.4

Rewrite the expression.

$f^{- 1} (\frac{5}{2} x - 5) = \frac{\frac{5 (x - 2)}{1}}{5} + 2$

Step 5.2.3.3.2

Divide $5 (x - 2)$ by $1$ .

$f^{- 1} (\frac{5}{2} x - 5) = \frac{5 (x - 2)}{5} + 2$

Step 5.2.3.4

Cancel the common factor of $5$ .

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Step 5.2.3.4.1

Cancel the common factor.

$f^{- 1} (\frac{5}{2} x - 5) = \frac{5 (x - 2)}{5} + 2$

Step 5.2.3.4.2

Divide $x - 2$ by $1$ .

$f^{- 1} (\frac{5}{2} x - 5) = x - 2 + 2$

Step 5.2.4

Combine the opposite terms in $x - 2 + 2$ .

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Step 5.2.4.1

Add $- 2$ and $2$ .

$f^{- 1} (\frac{5}{2} x - 5) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{5}{2} x - 5) = x$

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 $f (\frac{2 x}{5} + 2)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{2 x}{5} + 2) = \frac{5}{2} \cdot (\frac{2 x}{5} + 2) - 5$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (\frac{2 x}{5} + 2) = \frac{5}{2} \cdot \frac{2 x}{5} + \frac{5}{2} \cdot 2 - 5$

Step 5.3.3.2

Cancel the common factor of $5$ .

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Step 5.3.3.2.1

Cancel the common factor.

$f (\frac{2 x}{5} + 2) = \frac{5}{2} \cdot \frac{2 x}{5} + \frac{5}{2} \cdot 2 - 5$

Step 5.3.3.2.2

Rewrite the expression.

$f (\frac{2 x}{5} + 2) = \frac{1}{2} \cdot (2 x) + \frac{5}{2} \cdot 2 - 5$

Step 5.3.3.3

Cancel the common factor of $2$ .

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Step 5.3.3.3.1

Factor $2$ out of $2 x$ .

$f (\frac{2 x}{5} + 2) = \frac{1}{2} \cdot (2 (x)) + \frac{5}{2} \cdot 2 - 5$

Step 5.3.3.3.2

Cancel the common factor.

$f (\frac{2 x}{5} + 2) = \frac{1}{2} \cdot (2 x) + \frac{5}{2} \cdot 2 - 5$

Step 5.3.3.3.3

Rewrite the expression.

$f (\frac{2 x}{5} + 2) = x + \frac{5}{2} \cdot 2 - 5$

Step 5.3.3.4

Cancel the common factor of $2$ .

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Step 5.3.3.4.1

Cancel the common factor.

$f (\frac{2 x}{5} + 2) = x + \frac{5}{2} \cdot 2 - 5$

Step 5.3.3.4.2

Rewrite the expression.

$f (\frac{2 x}{5} + 2) = x + 5 - 5$

Step 5.3.4

Combine the opposite terms in $x + 5 - 5$ .

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Step 5.3.4.1

Subtract $5$ from $5$ .

$f (\frac{2 x}{5} + 2) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{2 x}{5} + 2) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{2 x}{5} + 2$ is the inverse of $f (x) = \frac{5}{2} x - 5$ .

$f^{- 1} (x) = \frac{2 x}{5} + 2$