Find the Inverse 8-x

$8 - x$

Step 1

Interchange the variables.

$x = 8 - y$

Step 2

Solve for $y$ .

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

$8 - y = x$

Subtract $8$ from both sides of the equation.

$- y = x - 8$

Divide each term in $- y = x - 8$ by $- 1$ and simplify.

Divide each term in $- y = x - 8$ by $- 1$ .

$\frac{- y}{- 1} = \frac{x}{- 1} + \frac{- 8}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{x}{- 1} + \frac{- 8}{- 1}$

Divide $y$ by $1$ .

$y = \frac{x}{- 1} + \frac{- 8}{- 1}$

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$y = - 1 \cdot x + \frac{- 8}{- 1}$

Rewrite $- 1 \cdot x$ as $- x$ .

$y = - x + \frac{- 8}{- 1}$

Divide $- 8$ by $- 1$ .

$y = - x + 8$

Step 3

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

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

Step 4

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

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

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

Simplify each term.

Apply the distributive property.

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

Multiply $- 1$ by $8$ .

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

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

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

Add $- 8$ and $8$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Apply the distributive property.

$f (- x + 8) = 8 + x - 1 \cdot 8$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$f (- x + 8) = 8 + 1 x - 1 \cdot 8$

Multiply $x$ by $1$ .

$f (- x + 8) = 8 + x - 1 \cdot 8$

Multiply $- 1$ by $8$ .

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

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

Subtract $8$ from $8$ .

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

Add $x$ and $0$ .

$f (- x + 8) = x$

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

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

Find the Inverse 8-x

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