Find the Inverse f(x)=x/8-7

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

Step 1

Write $f (x) = \frac{x}{8} - 7$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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

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

Add $7$ to both sides of the equation.

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

Multiply both sides of the equation by $8$ .

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

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

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

Rewrite the expression.

$y = 8 (x + 7)$

Simplify the right side.

Simplify $8 (x + 7)$ .

Apply the distributive property.

$y = 8 x + 8 \cdot 7$

Multiply $8$ by $7$ .

$y = 8 x + 56$

Step 4

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

$f^{- 1} (x) = 8 x + 56$

Step 5

Verify if $f^{- 1} (x) = 8 x + 56$ is the inverse of $f (x) = \frac{x}{8} - 7$ .

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

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

Set up the composite result function.

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

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

$f^{- 1} (\frac{x}{8} - 7) = 8 (\frac{x}{8} - 7) + 56$

Simplify each term.

Apply the distributive property.

$f^{- 1} (\frac{x}{8} - 7) = 8 (\frac{x}{8}) + 8 \cdot - 7 + 56$

Cancel the common factor of $8$ .

Cancel the common factor.

$f^{- 1} (\frac{x}{8} - 7) = 8 (\frac{x}{8}) + 8 \cdot - 7 + 56$

Rewrite the expression.

$f^{- 1} (\frac{x}{8} - 7) = x + 8 \cdot - 7 + 56$

Multiply $8$ by $- 7$ .

$f^{- 1} (\frac{x}{8} - 7) = x - 56 + 56$

Combine the opposite terms in $x - 56 + 56$ .

Add $- 56$ and $56$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

Evaluate $f (8 x + 56)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (8 x + 56) = \frac{8 x + 56}{8} - 7$

Cancel the common factor of $8 x + 56$ and $8$ .

Factor $8$ out of $8 x$ .

$f (8 x + 56) = \frac{8 (x) + 56}{8} - 7$

Factor $8$ out of $56$ .

$f (8 x + 56) = \frac{8 (x) + 8 \cdot 7}{8} - 7$

Factor $8$ out of $8 (x) + 8 (7)$ .

$f (8 x + 56) = \frac{8 (x + 7)}{8} - 7$

Cancel the common factors.

Factor $8$ out of $8$ .

$f (8 x + 56) = \frac{8 (x + 7)}{8 (1)} - 7$

Cancel the common factor.

$f (8 x + 56) = \frac{8 (x + 7)}{8 \cdot 1} - 7$

Rewrite the expression.

$f (8 x + 56) = \frac{x + 7}{1} - 7$

Divide $x + 7$ by $1$ .

$f (8 x + 56) = x + 7 - 7$

Combine the opposite terms in $x + 7 - 7$ .

Subtract $7$ from $7$ .

$f (8 x + 56) = x + 0$

Add $x$ and $0$ .

$f (8 x + 56) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 8 x + 56$ is the inverse of $f (x) = \frac{x}{8} - 7$ .

$f^{- 1} (x) = 8 x + 56$

Find the Inverse f(x)=x/8-7

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