Trigonometry Examples

$f (x) = \frac{x - 10}{x - 7}$

Step 1

Write $f (x) = \frac{x - 10}{x - 7}$ as an equation.

$y = \frac{x - 10}{x - 7}$

Step 2

Interchange the variables.

$x = \frac{y - 10}{y - 7}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{y - 10}{y - 7} = x$ .

$\frac{y - 10}{y - 7} = x$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y - 7, 1$

Step 3.2.2

Remove parentheses.

$y - 7, 1$

Step 3.2.3

The LCM of one and any expression is the expression.

$y - 7$

Step 3.3

Multiply each term in $\frac{y - 10}{y - 7} = x$ by $y - 7$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{y - 10}{y - 7} = x$ by $y - 7$ .

$\frac{y - 10}{y - 7} (y - 7) = x (y - 7)$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $y - 7$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{y - 10}{y - 7} (y - 7) = x (y - 7)$

Step 3.3.2.1.2

Rewrite the expression.

$y - 10 = x (y - 7)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Apply the distributive property.

$y - 10 = x y + x \cdot - 7$

Step 3.3.3.2

Move $- 7$ to the left of $x$ .

$y - 10 = x y - 7 x$

Step 3.4

Solve the equation.

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Step 3.4.1

Subtract $x y$ from both sides of the equation.

$y - 10 - x y = - 7 x$

Step 3.4.2

Add $10$ to both sides of the equation.

$y - x y = - 7 x + 10$

Step 3.4.3

Factor $y$ out of $y - x y$ .

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Step 3.4.3.1

Factor $y$ out of $y^{1}$ .

$y \cdot 1 - x y = - 7 x + 10$

Step 3.4.3.2

Factor $y$ out of $- x y$ .

$y \cdot 1 + y (- x) = - 7 x + 10$

Step 3.4.3.3

Factor $y$ out of $y \cdot 1 + y (- x)$ .

$y (1 - x) = - 7 x + 10$

Step 3.4.4

Divide each term in $y (1 - x) = - 7 x + 10$ by $1 - x$ and simplify.

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Step 3.4.4.1

Divide each term in $y (1 - x) = - 7 x + 10$ by $1 - x$ .

$\frac{y (1 - x)}{1 - x} = \frac{- 7 x}{1 - x} + \frac{10}{1 - x}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $1 - x$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y (1 - x)}{1 - x} = \frac{- 7 x}{1 - x} + \frac{10}{1 - x}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 7 x}{1 - x} + \frac{10}{1 - x}$

Step 3.4.4.3

Simplify the right side.

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Step 3.4.4.3.1

Combine the numerators over the common denominator.

$y = \frac{- 7 x + 10}{1 - x}$

Step 3.4.4.3.2

Factor $- 1$ out of $- 7 x$ .

$y = \frac{- (7 x) + 10}{1 - x}$

Step 3.4.4.3.3

Rewrite $10$ as $- 1 (- 10)$ .

$y = \frac{- (7 x) - 1 (- 10)}{1 - x}$

Step 3.4.4.3.4

Factor $- 1$ out of $- (7 x) - 1 (- 10)$ .

$y = \frac{- (7 x - 10)}{1 - x}$

Step 3.4.4.3.5

Simplify the expression.

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Step 3.4.4.3.5.1

Rewrite $- (7 x - 10)$ as $- 1 (7 x - 10)$ .

$y = \frac{- 1 (7 x - 10)}{1 - x}$

Step 3.4.4.3.5.2

Move the negative in front of the fraction.

$y = - \frac{7 x - 10}{1 - x}$

Step 4

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

$f^{- 1} (x) = - \frac{7 x - 10}{1 - x}$

Step 5

Verify if $f^{- 1} (x) = - \frac{7 x - 10}{1 - x}$ is the inverse of $f (x) = \frac{x - 10}{x - 7}$ .

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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 $f^{- 1} (\frac{x - 10}{x - 7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{7 (\frac{x - 10}{x - 7}) - 10}{1 - (\frac{x - 10}{x - 7})}$

Step 5.2.3

Multiply the numerator and denominator of the fraction by $x - 7$ .

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Step 5.2.3.1

Multiply $\frac{7 \frac{x - 10}{x - 7} - 10}{1 - \frac{x - 10}{x - 7}}$ by $\frac{x - 7}{x - 7}$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - (\frac{x - 7}{x - 7} \cdot \frac{7 (\frac{x - 10}{x - 7}) - 10}{1 - \frac{x - 10}{x - 7}})$

Step 5.2.3.2

Combine.

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (7 (\frac{x - 10}{x - 7}) - 10)}{(x - 7) (1 - \frac{x - 10}{x - 7})}$

Step 5.2.4

Apply the distributive property.

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (7 (\frac{x - 10}{x - 7})) + (x - 7) \cdot - 10}{(x - 7) \cdot 1 + (x - 7) (- \frac{x - 10}{x - 7})}$

Step 5.2.5

Cancel the common factor of $x - 7$ .

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Step 5.2.5.1

Move the leading negative in $- \frac{x - 10}{x - 7}$ into the numerator.

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (7 (\frac{x - 10}{x - 7})) + (x - 7) \cdot - 10}{(x - 7) \cdot 1 + (x - 7) (\frac{- (x - 10)}{x - 7})}$

Step 5.2.5.2

Cancel the common factor.

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (7 (\frac{x - 10}{x - 7})) + (x - 7) \cdot - 10}{(x - 7) \cdot 1 + (x - 7) (\frac{- (x - 10)}{x - 7})}$

Step 5.2.5.3

Rewrite the expression.

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (7 (\frac{x - 10}{x - 7})) + (x - 7) \cdot - 10}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6

Simplify the numerator.

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Step 5.2.6.1

Factor $x - 7$ out of $(x - 7) \cdot 7 \frac{x - 10}{x - 7} + (x - 7) \cdot - 10$ .

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Step 5.2.6.1.1

Factor $x - 7$ out of $(x - 7) \cdot 7 \frac{x - 10}{x - 7}$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (7 (\frac{x - 10}{x - 7})) + (x - 7) \cdot - 10}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6.1.2

Factor $x - 7$ out of $(x - 7) \cdot - 10$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (7 (\frac{x - 10}{x - 7})) + (x - 7) (- 10)}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6.1.3

Factor $x - 7$ out of $(x - 7) (7 \frac{x - 10}{x - 7}) + (x - 7) (- 10)$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (7 (\frac{x - 10}{x - 7}) - 10)}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6.2

Combine $7$ and $\frac{x - 10}{x - 7}$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (\frac{7 (x - 10)}{x - 7} - 10)}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6.3

To write $- 10$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (\frac{7 (x - 10)}{x - 7} - 10 \cdot \frac{x - 7}{x - 7})}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6.4

Combine $- 10$ and $\frac{x - 7}{x - 7}$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (\frac{7 (x - 10)}{x - 7} + \frac{- 10 (x - 7)}{x - 7})}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6.5

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (\frac{7 (x - 10) - 10 (x - 7)}{x - 7})}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6.6

Rewrite $\frac{7 (x - 10) - 10 (x - 7)}{x - 7}$ in a factored form.

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Step 5.2.6.6.1

Apply the distributive property.

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (\frac{7 x + 7 \cdot - 10 - 10 (x - 7)}{x - 7})}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6.6.2

Multiply $7$ by $- 10$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (\frac{7 x - 70 - 10 (x - 7)}{x - 7})}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6.6.3

Apply the distributive property.

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (\frac{7 x - 70 - 10 x - 10 \cdot - 7}{x - 7})}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6.6.4

Multiply $- 10$ by $- 7$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (\frac{7 x - 70 - 10 x + 70}{x - 7})}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6.6.5

Subtract $10 x$ from $7 x$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (\frac{- 3 x - 70 + 70}{x - 7})}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6.6.6

Add $- 70$ and $70$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (\frac{- 3 x + 0}{x - 7})}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6.6.7

Add $- 3 x$ and $0$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (\frac{- 3 x}{x - 7})}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6.7

Move the negative in front of the fraction.

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) (- \frac{(3) x}{x - 7})}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6.8

Remove unnecessary parentheses.

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{(x - 7) \cdot (- 1 \frac{3 x}{x - 7})}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.6.9

Factor out negative.

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{- (x - 7) \frac{3 x}{x - 7}}{(x - 7) \cdot 1 - (x - 10)}$

Step 5.2.7

Simplify the denominator.

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Step 5.2.7.1

Multiply $x - 7$ by $1$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{- (x - 7) \frac{3 x}{x - 7}}{x - 7 - (x - 10)}$

Step 5.2.7.2

Apply the distributive property.

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{- (x - 7) \frac{3 x}{x - 7}}{x - 7 - x + 10}$

Step 5.2.7.3

Multiply $- 1$ by $- 10$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{- (x - 7) \frac{3 x}{x - 7}}{x - 7 - x + 10}$

Step 5.2.7.4

Subtract $x$ from $x$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{- (x - 7) \frac{3 x}{x - 7}}{0 - 7 + 10}$

Step 5.2.7.5

Subtract $7$ from $0$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{- (x - 7) \frac{3 x}{x - 7}}{- 7 + 10}$

Step 5.2.7.6

Add $- 7$ and $10$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - \frac{- (x - 7) \frac{3 x}{x - 7}}{3}$

Step 5.2.8

Reduce the expression by cancelling the common factors.

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Step 5.2.8.1

Factor $\frac{3 x}{x - 7}$ out of $\frac{- (x - 7) \frac{3 x}{x - 7}}{3}$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - (\frac{3 x}{x - 7} \cdot \frac{- (x - 7)}{3})$

Step 5.2.8.2

Cancel the common factor of $3$ .

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Step 5.2.8.2.1

Factor $3$ out of $3 x$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - (\frac{3 (x)}{x - 7} \cdot \frac{- (x - 7)}{3})$

Step 5.2.8.2.2

Cancel the common factor.

$f^{- 1} (\frac{x - 10}{x - 7}) = - (\frac{3 x}{x - 7} \cdot \frac{- (x - 7)}{3})$

Step 5.2.8.2.3

Rewrite the expression.

$f^{- 1} (\frac{x - 10}{x - 7}) = - (\frac{x}{x - 7} \cdot (- (x - 7)))$

Step 5.2.8.3

Cancel the common factor of $x - 7$ .

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Step 5.2.8.3.1

Factor $x - 7$ out of $- (x - 7)$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - (\frac{x}{x - 7} \cdot ((x - 7) \cdot - 1))$

Step 5.2.8.3.2

Cancel the common factor.

$f^{- 1} (\frac{x - 10}{x - 7}) = - (\frac{x}{x - 7} \cdot ((x - 7) \cdot - 1))$

Step 5.2.8.3.3

Rewrite the expression.

$f^{- 1} (\frac{x - 10}{x - 7}) = - (x \cdot - 1)$

Step 5.2.8.4

Simplify the expression.

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Step 5.2.8.4.1

Move $- 1$ to the left of $x$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = - (- 1 \cdot x)$

Step 5.2.8.4.2

Rewrite $- 1 x$ as $- x$ .

$f^{- 1} (\frac{x - 10}{x - 7}) = x$

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 $f (- \frac{7 x - 10}{1 - x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{7 x - 10}{1 - x}) = \frac{(- \frac{7 x - 10}{1 - x}) - 10}{(- \frac{7 x - 10}{1 - x}) - 7}$

Step 5.3.3

Multiply the numerator and denominator of the fraction by $1 - x$ .

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Step 5.3.3.1

Multiply $\frac{- \frac{7 x - 10}{1 - x} - 10}{- \frac{7 x - 10}{1 - x} - 7}$ by $\frac{1 - x}{1 - x}$ .

$f (- \frac{7 x - 10}{1 - x}) = \frac{1 - x}{1 - x} \cdot \frac{- \frac{7 x - 10}{1 - x} - 10}{- \frac{7 x - 10}{1 - x} - 7}$

Step 5.3.3.2

Combine.

$f (- \frac{7 x - 10}{1 - x}) = \frac{(1 - x) (- \frac{7 x - 10}{1 - x} - 10)}{(1 - x) (- \frac{7 x - 10}{1 - x} - 7)}$

Step 5.3.4

Apply the distributive property.

$f (- \frac{7 x - 10}{1 - x}) = \frac{(1 - x) (- \frac{7 x - 10}{1 - x}) + (1 - x) \cdot - 10}{(1 - x) (- \frac{7 x - 10}{1 - x}) + (1 - x) \cdot - 7}$

Step 5.3.5

Simplify by cancelling.

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Step 5.3.5.1

Cancel the common factor of $1 - x$ .

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Step 5.3.5.1.1

Move the leading negative in $- \frac{7 x - 10}{1 - x}$ into the numerator.

$f (- \frac{7 x - 10}{1 - x}) = \frac{(1 - x) (\frac{- (7 x - 10)}{1 - x}) + (1 - x) \cdot - 10}{(1 - x) (- \frac{7 x - 10}{1 - x}) + (1 - x) \cdot - 7}$

Step 5.3.5.1.2

Cancel the common factor.

$f (- \frac{7 x - 10}{1 - x}) = \frac{(1 - x) (\frac{- (7 x - 10)}{1 - x}) + (1 - x) \cdot - 10}{(1 - x) (- \frac{7 x - 10}{1 - x}) + (1 - x) \cdot - 7}$

Step 5.3.5.1.3

Rewrite the expression.

$f (- \frac{7 x - 10}{1 - x}) = \frac{- (7 x - 10) + (1 - x) \cdot - 10}{(1 - x) (- \frac{7 x - 10}{1 - x}) + (1 - x) \cdot - 7}$

Step 5.3.5.2

Cancel the common factor of $1 - x$ .

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Step 5.3.5.2.1

Move the leading negative in $- \frac{7 x - 10}{1 - x}$ into the numerator.

$f (- \frac{7 x - 10}{1 - x}) = \frac{- (7 x - 10) + (1 - x) \cdot - 10}{(1 - x) (\frac{- (7 x - 10)}{1 - x}) + (1 - x) \cdot - 7}$

Step 5.3.5.2.2

Cancel the common factor.

$f (- \frac{7 x - 10}{1 - x}) = \frac{- (7 x - 10) + (1 - x) \cdot - 10}{(1 - x) (\frac{- (7 x - 10)}{1 - x}) + (1 - x) \cdot - 7}$

Step 5.3.5.2.3

Rewrite the expression.

$f (- \frac{7 x - 10}{1 - x}) = \frac{- (7 x - 10) + (1 - x) \cdot - 10}{- (7 x - 10) + (1 - x) \cdot - 7}$

Step 5.3.6

Simplify the numerator.

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Step 5.3.6.1

Apply the distributive property.

$f (- \frac{7 x - 10}{1 - x}) = \frac{- (7 x) + 10 + (1 - x) \cdot - 10}{- (7 x - 10) + (1 - x) \cdot - 7}$

Step 5.3.6.2

Multiply $7$ by $- 1$ .

$f (- \frac{7 x - 10}{1 - x}) = \frac{- 7 x + 10 + (1 - x) \cdot - 10}{- (7 x - 10) + (1 - x) \cdot - 7}$

Step 5.3.6.3

Multiply $- 1$ by $- 10$ .

$f (- \frac{7 x - 10}{1 - x}) = \frac{- 7 x + 10 + (1 - x) \cdot - 10}{- (7 x - 10) + (1 - x) \cdot - 7}$

Step 5.3.6.4

Apply the distributive property.

$f (- \frac{7 x - 10}{1 - x}) = \frac{- 7 x + 10 + 1 \cdot - 10 - x \cdot - 10}{- (7 x - 10) + (1 - x) \cdot - 7}$

Step 5.3.6.5

Multiply $- 10$ by $1$ .

$f (- \frac{7 x - 10}{1 - x}) = \frac{- 7 x + 10 - 10 - x \cdot - 10}{- (7 x - 10) + (1 - x) \cdot - 7}$

Step 5.3.6.6

Multiply $- 10$ by $- 1$ .

$f (- \frac{7 x - 10}{1 - x}) = \frac{- 7 x + 10 - 10 + 10 x}{- (7 x - 10) + (1 - x) \cdot - 7}$

Step 5.3.6.7

Add $- 7 x$ and $10 x$ .

$f (- \frac{7 x - 10}{1 - x}) = \frac{3 x + 10 - 10}{- (7 x - 10) + (1 - x) \cdot - 7}$

Step 5.3.6.8

Subtract $10$ from $10$ .

$f (- \frac{7 x - 10}{1 - x}) = \frac{3 x + 0}{- (7 x - 10) + (1 - x) \cdot - 7}$

Step 5.3.6.9

Add $3 x$ and $0$ .

$f (- \frac{7 x - 10}{1 - x}) = \frac{3 x}{- (7 x - 10) + (1 - x) \cdot - 7}$

Step 5.3.7

Simplify the denominator.

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Step 5.3.7.1

Apply the distributive property.

$f (- \frac{7 x - 10}{1 - x}) = \frac{3 x}{- (7 x) + 10 + (1 - x) \cdot - 7}$

Step 5.3.7.2

Multiply $7$ by $- 1$ .

$f (- \frac{7 x - 10}{1 - x}) = \frac{3 x}{- 7 x + 10 + (1 - x) \cdot - 7}$

Step 5.3.7.3

Multiply $- 1$ by $- 10$ .

$f (- \frac{7 x - 10}{1 - x}) = \frac{3 x}{- 7 x + 10 + (1 - x) \cdot - 7}$

Step 5.3.7.4

Apply the distributive property.

$f (- \frac{7 x - 10}{1 - x}) = \frac{3 x}{- 7 x + 10 + 1 \cdot - 7 - x \cdot - 7}$

Step 5.3.7.5

Multiply $- 7$ by $1$ .

$f (- \frac{7 x - 10}{1 - x}) = \frac{3 x}{- 7 x + 10 - 7 - x \cdot - 7}$

Step 5.3.7.6

Multiply $- 7$ by $- 1$ .

$f (- \frac{7 x - 10}{1 - x}) = \frac{3 x}{- 7 x + 10 - 7 + 7 x}$

Step 5.3.7.7

Add $- 7 x$ and $7 x$ .

$f (- \frac{7 x - 10}{1 - x}) = \frac{3 x}{0 + 10 - 7}$

Step 5.3.7.8

Add $0$ and $10$ .

$f (- \frac{7 x - 10}{1 - x}) = \frac{3 x}{10 - 7}$

Step 5.3.7.9

Subtract $7$ from $10$ .

$f (- \frac{7 x - 10}{1 - x}) = \frac{3 x}{3}$

Step 5.3.8

Cancel the common factor of $3$ .

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Step 5.3.8.1

Cancel the common factor.

$f (- \frac{7 x - 10}{1 - x}) = \frac{3 x}{3}$

Step 5.3.8.2

Divide $x$ by $1$ .

$f (- \frac{7 x - 10}{1 - x}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{7 x - 10}{1 - x}$ is the inverse of $f (x) = \frac{x - 10}{x - 7}$ .

$f^{- 1} (x) = - \frac{7 x - 10}{1 - x}$