Trigonometry Examples

$f (x) = - 10 x$

Step 1

Write $f (x) = - 10 x$ as an equation.

$y = - 10 x$

Step 2

Interchange the variables.

$x = - 10 y$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- 10 y = x$ .

$- 10 y = x$

Step 3.2

Divide each term in $- 10 y = x$ by $- 10$ and simplify.

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Step 3.2.1

Divide each term in $- 10 y = x$ by $- 10$ .

$\frac{- 10 y}{- 10} = \frac{x}{- 10}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 10$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 10 y}{- 10} = \frac{x}{- 10}$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{- 10}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

$y = - \frac{x}{10}$

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = - \frac{x}{10}$ is the inverse of $f (x) = - 10 x$ .

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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 $f^{- 1} (- 10 x)$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 5.2.3

Cancel the common factor of $- 10$ and $10$ .

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Step 5.2.3.1

Factor $10$ out of $- 10 x$ .

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

Step 5.2.3.2

Cancel the common factors.

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Step 5.2.3.2.1

Factor $10$ out of $10$ .

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

Step 5.2.3.2.2

Cancel the common factor.

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

Step 5.2.3.2.3

Rewrite the expression.

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

Step 5.2.3.2.4

Divide $- x$ by $1$ .

$f^{- 1} (- 10 x) = x$

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 $f (- \frac{x}{10})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{x}{10}) = - 10 (- \frac{x}{10})$

Step 5.3.3

Cancel the common factor of $10$ .

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Step 5.3.3.1

Move the leading negative in $- \frac{x}{10}$ into the numerator.

$f (- \frac{x}{10}) = - 10 \frac{- x}{10}$

Step 5.3.3.2

Factor $10$ out of $- 10$ .

$f (- \frac{x}{10}) = 10 (- 1) (\frac{- x}{10})$

Step 5.3.3.3

Cancel the common factor.

$f (- \frac{x}{10}) = 10 \cdot (- 1 \frac{- x}{10})$

Step 5.3.3.4

Rewrite the expression.

$f (- \frac{x}{10}) = - 1 (- x)$

Step 5.3.4

Multiply.

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Step 5.3.4.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{x}{10}) = 1 x$

Step 5.3.4.2

Multiply $x$ by $1$ .

$f (- \frac{x}{10}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{x}{10}$ is the inverse of $f (x) = - 10 x$ .

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