Trigonometry Examples

$y = \frac{x - 8}{7}$

Step 1

Interchange the variables.

$x = \frac{y - 8}{7}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{y - 8}{7} = x$ .

$\frac{y - 8}{7} = x$

Step 2.2

Multiply both sides of the equation by $7$ .

$7 \frac{y - 8}{7} = 7 x$

Step 2.3

Simplify the left side.

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Step 2.3.1

Cancel the common factor of $7$ .

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Step 2.3.1.1

Cancel the common factor.

$7 \frac{y - 8}{7} = 7 x$

Step 2.3.1.2

Rewrite the expression.

$y - 8 = 7 x$

Step 2.4

Add $8$ to both sides of the equation.

$y = 7 x + 8$

Step 3

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

$f^{- 1} (x) = 7 x + 8$

Step 4

Verify if $f^{- 1} (x) = 7 x + 8$ is the inverse of $f (x) = \frac{x - 8}{7}$ .

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

Evaluate $f^{- 1} (f (x))$ .

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (\frac{x - 8}{7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x - 8}{7}) = 7 (\frac{x - 8}{7}) + 8$

Step 4.2.3

Cancel the common factor of $7$ .

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Step 4.2.3.1

Cancel the common factor.

$f^{- 1} (\frac{x - 8}{7}) = 7 (\frac{x - 8}{7}) + 8$

Step 4.2.3.2

Rewrite the expression.

$f^{- 1} (\frac{x - 8}{7}) = x - 8 + 8$

Step 4.2.4

Combine the opposite terms in $x - 8 + 8$ .

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Step 4.2.4.1

Add $- 8$ and $8$ .

$f^{- 1} (\frac{x - 8}{7}) = x + 0$

Step 4.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{x - 8}{7}) = x$

Step 4.3

Evaluate $f (f^{- 1} (x))$ .

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (7 x + 8)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (7 x + 8) = \frac{(7 x + 8) - 8}{7}$

Step 4.3.3

Simplify the numerator.

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Step 4.3.3.1

Subtract $8$ from $8$ .

$f (7 x + 8) = \frac{7 x + 0}{7}$

Step 4.3.3.2

Add $7 x$ and $0$ .

$f (7 x + 8) = \frac{7 x}{7}$

Step 4.3.4

Cancel the common factor of $7$ .

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Step 4.3.4.1

Cancel the common factor.

$f (7 x + 8) = \frac{7 x}{7}$

Step 4.3.4.2

Divide $x$ by $1$ .

$f (7 x + 8) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 7 x + 8$ is the inverse of $f (x) = \frac{x - 8}{7}$ .

$f^{- 1} (x) = 7 x + 8$