Find the Inverse y=-8x+16

$y = - 8 x + 16$

Step 1

Interchange the variables.

$x = - 8 y + 16$

Step 2

Solve for $y$ .

Rewrite the equation as $- 8 y + 16 = x$ .

$- 8 y + 16 = x$

Subtract $16$ from both sides of the equation.

$- 8 y = x - 16$

Divide each term in $- 8 y = x - 16$ by $- 8$ and simplify.

Divide each term in $- 8 y = x - 16$ by $- 8$ .

$\frac{- 8 y}{- 8} = \frac{x}{- 8} + \frac{- 16}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 y}{- 8} = \frac{x}{- 8} + \frac{- 16}{- 8}$

Divide $y$ by $1$ .

$y = \frac{x}{- 8} + \frac{- 16}{- 8}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{x}{8} + \frac{- 16}{- 8}$

Divide $- 16$ by $- 8$ .

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

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = - \frac{x}{8} + 2$ is the inverse of $f (x) = - 8 x + 16$ .

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 $f^{- 1} (- 8 x + 16)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- 8 x + 16) = - \frac{- 8 x + 16}{8} + 2$

Simplify each term.

Cancel the common factor of $- 8 x + 16$ and $8$ .

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

$f^{- 1} (- 8 x + 16) = - \frac{8 (- x) + 16}{8} + 2$

Factor $8$ out of $16$ .

$f^{- 1} (- 8 x + 16) = - \frac{8 (- x) + 8 \cdot 2}{8} + 2$

Factor $8$ out of $8 (- x) + 8 (2)$ .

$f^{- 1} (- 8 x + 16) = - \frac{8 (- x + 2)}{8} + 2$

Cancel the common factors.

Factor $8$ out of $8$ .

$f^{- 1} (- 8 x + 16) = - \frac{8 (- x + 2)}{8 (1)} + 2$

Cancel the common factor.

$f^{- 1} (- 8 x + 16) = - \frac{8 (- x + 2)}{8 \cdot 1} + 2$

Rewrite the expression.

$f^{- 1} (- 8 x + 16) = - \frac{- x + 2}{1} + 2$

Divide $- x + 2$ by $1$ .

$f^{- 1} (- 8 x + 16) = - (- x + 2) + 2$

Apply the distributive property.

$f^{- 1} (- 8 x + 16) = x - 1 \cdot 2 + 2$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- 8 x + 16) = 1 x - 1 \cdot 2 + 2$

Multiply $x$ by $1$ .

$f^{- 1} (- 8 x + 16) = x - 1 \cdot 2 + 2$

Multiply $- 1$ by $2$ .

$f^{- 1} (- 8 x + 16) = x - 2 + 2$

Combine the opposite terms in $x - 2 + 2$ .

Add $- 2$ and $2$ .

$f^{- 1} (- 8 x + 16) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (- 8 x + 16) = x$

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

Set up the composite result function.

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

Evaluate $f (- \frac{x}{8} + 2)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{x}{8} + 2) = - 8 (- \frac{x}{8} + 2) + 16$

Simplify each term.

Apply the distributive property.

$f (- \frac{x}{8} + 2) = - 8 (- \frac{x}{8}) - 8 \cdot 2 + 16$

Cancel the common factor of $8$ .

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

$f (- \frac{x}{8} + 2) = - 8 \frac{- x}{8} - 8 \cdot 2 + 16$

Factor $8$ out of $- 8$ .

$f (- \frac{x}{8} + 2) = 8 (- 1) (\frac{- x}{8}) - 8 \cdot 2 + 16$

Cancel the common factor.

$f (- \frac{x}{8} + 2) = 8 \cdot (- 1 \frac{- x}{8}) - 8 \cdot 2 + 16$

Rewrite the expression.

$f (- \frac{x}{8} + 2) = - 1 (- x) - 8 \cdot 2 + 16$

Multiply $- 1$ by $- 1$ .

$f (- \frac{x}{8} + 2) = 1 x - 8 \cdot 2 + 16$

Multiply $x$ by $1$ .

$f (- \frac{x}{8} + 2) = x - 8 \cdot 2 + 16$

Multiply $- 8$ by $2$ .

$f (- \frac{x}{8} + 2) = x - 16 + 16$

Combine the opposite terms in $x - 16 + 16$ .

Add $- 16$ and $16$ .

$f (- \frac{x}{8} + 2) = x + 0$

Add $x$ and $0$ .

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

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{x}{8} + 2$ is the inverse of $f (x) = - 8 x + 16$ .

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

Find the Inverse y=-8x+16

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