Trigonometry Examples

$f (x) = - 8 x - 6$

Step 1

Write $f (x) = - 8 x - 6$ as an equation.

$y = - 8 x - 6$

Step 2

Interchange the variables.

$x = - 8 y - 6$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- 8 y - 6 = x$ .

$- 8 y - 6 = x$

Step 3.2

Add $6$ to both sides of the equation.

$- 8 y = x + 6$

Step 3.3

Divide each term in $- 8 y = x + 6$ by $- 8$ and simplify.

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Step 3.3.1

Divide each term in $- 8 y = x + 6$ by $- 8$ .

$\frac{- 8 y}{- 8} = \frac{x}{- 8} + \frac{6}{- 8}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 8$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 8 y}{- 8} = \frac{x}{- 8} + \frac{6}{- 8}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{- 8} + \frac{6}{- 8}$

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

Move the negative in front of the fraction.

$y = - \frac{x}{8} + \frac{6}{- 8}$

Step 3.3.3.1.2

Cancel the common factor of $6$ and $- 8$ .

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Step 3.3.3.1.2.1

Factor $2$ out of $6$ .

$y = - \frac{x}{8} + \frac{2 (3)}{- 8}$

Step 3.3.3.1.2.2

Cancel the common factors.

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Step 3.3.3.1.2.2.1

Factor $2$ out of $- 8$ .

$y = - \frac{x}{8} + \frac{2 \cdot 3}{2 \cdot - 4}$

Step 3.3.3.1.2.2.2

Cancel the common factor.

$y = - \frac{x}{8} + \frac{2 \cdot 3}{2 \cdot - 4}$

Step 3.3.3.1.2.2.3

Rewrite the expression.

$y = - \frac{x}{8} + \frac{3}{- 4}$

Step 3.3.3.1.3

Move the negative in front of the fraction.

$y = - \frac{x}{8} - \frac{3}{4}$

Step 4

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

$f^{- 1} (x) = - \frac{x}{8} - \frac{3}{4}$

Step 5

Verify if $f^{- 1} (x) = - \frac{x}{8} - \frac{3}{4}$ is the inverse of $f (x) = - 8 x - 6$ .

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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 $f^{- 1} (- 8 x - 6)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- 8 x - 6) = - \frac{- 8 x - 6}{8} - \frac{3}{4}$

Step 5.2.3

Simplify terms.

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Step 5.2.3.1

Cancel the common factor of $- 8 x - 6$ and $8$ .

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Step 5.2.3.1.1

Factor $2$ out of $- 8 x$ .

$f^{- 1} (- 8 x - 6) = - \frac{2 (- 4 x) - 6}{8} - \frac{3}{4}$

Step 5.2.3.1.2

Factor $2$ out of $- 6$ .

$f^{- 1} (- 8 x - 6) = - \frac{2 (- 4 x) + 2 (- 3)}{8} - \frac{3}{4}$

Step 5.2.3.1.3

Factor $2$ out of $2 (- 4 x) + 2 (- 3)$ .

$f^{- 1} (- 8 x - 6) = - \frac{2 (- 4 x - 3)}{8} - \frac{3}{4}$

Step 5.2.3.1.4

Cancel the common factors.

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Step 5.2.3.1.4.1

Factor $2$ out of $8$ .

$f^{- 1} (- 8 x - 6) = - \frac{2 (- 4 x - 3)}{2 (4)} - \frac{3}{4}$

Step 5.2.3.1.4.2

Cancel the common factor.

$f^{- 1} (- 8 x - 6) = - \frac{2 (- 4 x - 3)}{2 \cdot 4} - \frac{3}{4}$

Step 5.2.3.1.4.3

Rewrite the expression.

$f^{- 1} (- 8 x - 6) = - \frac{- 4 x - 3}{4} - \frac{3}{4}$

Step 5.2.3.2

Combine the numerators over the common denominator.

$f^{- 1} (- 8 x - 6) = \frac{- (- 4 x - 3) - 3}{4}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Apply the distributive property.

$f^{- 1} (- 8 x - 6) = \frac{- (- 4 x) + 3 - 3}{4}$

Step 5.2.4.2

Multiply $- 4$ by $- 1$ .

$f^{- 1} (- 8 x - 6) = \frac{4 x + 3 - 3}{4}$

Step 5.2.4.3

Multiply $- 1$ by $- 3$ .

$f^{- 1} (- 8 x - 6) = \frac{4 x + 3 - 3}{4}$

Step 5.2.5

Simplify terms.

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Step 5.2.5.1

Combine the opposite terms in $4 x + 3 - 3$ .

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Step 5.2.5.1.1

Subtract $3$ from $3$ .

$f^{- 1} (- 8 x - 6) = \frac{4 x + 0}{4}$

Step 5.2.5.1.2

Add $4 x$ and $0$ .

$f^{- 1} (- 8 x - 6) = \frac{4 x}{4}$

Step 5.2.5.2

Cancel the common factor of $4$ .

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Step 5.2.5.2.1

Cancel the common factor.

$f^{- 1} (- 8 x - 6) = \frac{4 x}{4}$

Step 5.2.5.2.2

Divide $x$ by $1$ .

$f^{- 1} (- 8 x - 6) = x$

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 $f (- \frac{x}{8} - \frac{3}{4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{x}{8} - \frac{3}{4}) = - 8 (- \frac{x}{8} - \frac{3}{4}) - 6$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (- \frac{x}{8} - \frac{3}{4}) = - 8 (- \frac{x}{8}) - 8 (- \frac{3}{4}) - 6$

Step 5.3.3.2

Cancel the common factor of $8$ .

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Step 5.3.3.2.1

Move the leading negative in $- \frac{x}{8}$ into the numerator.

$f (- \frac{x}{8} - \frac{3}{4}) = - 8 \frac{- x}{8} - 8 (- \frac{3}{4}) - 6$

Step 5.3.3.2.2

Factor $8$ out of $- 8$ .

$f (- \frac{x}{8} - \frac{3}{4}) = 8 (- 1) (\frac{- x}{8}) - 8 (- \frac{3}{4}) - 6$

Step 5.3.3.2.3

Cancel the common factor.

$f (- \frac{x}{8} - \frac{3}{4}) = 8 \cdot (- 1 \frac{- x}{8}) - 8 (- \frac{3}{4}) - 6$

Step 5.3.3.2.4

Rewrite the expression.

$f (- \frac{x}{8} - \frac{3}{4}) = - 1 (- x) - 8 (- \frac{3}{4}) - 6$

Step 5.3.3.3

Multiply $- 1$ by $- 1$ .

$f (- \frac{x}{8} - \frac{3}{4}) = 1 x - 8 (- \frac{3}{4}) - 6$

Step 5.3.3.4

Multiply $x$ by $1$ .

$f (- \frac{x}{8} - \frac{3}{4}) = x - 8 (- \frac{3}{4}) - 6$

Step 5.3.3.5

Cancel the common factor of $4$ .

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Step 5.3.3.5.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$f (- \frac{x}{8} - \frac{3}{4}) = x - 8 (\frac{- 3}{4}) - 6$

Step 5.3.3.5.2

Factor $4$ out of $- 8$ .

$f (- \frac{x}{8} - \frac{3}{4}) = x + 4 (- 2) (\frac{- 3}{4}) - 6$

Step 5.3.3.5.3

Cancel the common factor.

$f (- \frac{x}{8} - \frac{3}{4}) = x + 4 \cdot (- 2 (\frac{- 3}{4})) - 6$

Step 5.3.3.5.4

Rewrite the expression.

$f (- \frac{x}{8} - \frac{3}{4}) = x - 2 \cdot - 3 - 6$

Step 5.3.3.6

Multiply $- 2$ by $- 3$ .

$f (- \frac{x}{8} - \frac{3}{4}) = x + 6 - 6$

Step 5.3.4

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

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Step 5.3.4.1

Subtract $6$ from $6$ .

$f (- \frac{x}{8} - \frac{3}{4}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (- \frac{x}{8} - \frac{3}{4}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{x}{8} - \frac{3}{4}$ is the inverse of $f (x) = - 8 x - 6$ .

$f^{- 1} (x) = - \frac{x}{8} - \frac{3}{4}$