Find the Inverse f(x)=(-16+x)/4

$f (x) = \frac{- 16 + x}{4}$

Step 1

Write $f (x) = \frac{- 16 + x}{4}$ as an equation.

$y = \frac{- 16 + x}{4}$

Step 2

Interchange the variables.

$x = \frac{- 16 + y}{4}$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{- 16 + y}{4} = x$ .

$\frac{- 16 + y}{4} = x$

Multiply both sides of the equation by $4$ .

$4 \frac{- 16 + y}{4} = 4 x$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$4 \frac{- 16 + y}{4} = 4 x$

Rewrite the expression.

$- 16 + y = 4 x$

Add $16$ to both sides of the equation.

$y = 4 x + 16$

Step 4

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

$f^{- 1} (x) = 4 x + 16$

Step 5

Verify if $f^{- 1} (x) = 4 x + 16$ is the inverse of $f (x) = \frac{- 16 + x}{4}$ .

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 $f^{- 1} (\frac{- 16 + x}{4})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{- 16 + x}{4}) = 4 (\frac{- 16 + x}{4}) + 16$

Cancel the common factor of $4$ .

Cancel the common factor.

$f^{- 1} (\frac{- 16 + x}{4}) = 4 (\frac{- 16 + x}{4}) + 16$

Rewrite the expression.

$f^{- 1} (\frac{- 16 + x}{4}) = - 16 + x + 16$

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

Add $- 16$ and $16$ .

$f^{- 1} (\frac{- 16 + x}{4}) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{- 16 + x}{4}) = x$

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

Set up the composite result function.

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

Evaluate $f (4 x + 16)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (4 x + 16) = \frac{- 16 + 4 x + 16}{4}$

Remove parentheses.

$f (4 x + 16) = \frac{- 16 + 4 x + 16}{4}$

Cancel the common factor of $- 16 + 4 x + 16$ and $4$ .

Factor $4$ out of $- 16$ .

$f (4 x + 16) = \frac{4 \cdot - 4 + 4 x + 16}{4}$

Factor $4$ out of $4 x$ .

$f (4 x + 16) = \frac{4 \cdot - 4 + 4 (x) + 16}{4}$

Factor $4$ out of $16$ .

$f (4 x + 16) = \frac{4 \cdot - 4 + 4 (x) + 4 \cdot 4}{4}$

Factor $4$ out of $4 (x) + 4 (4)$ .

$f (4 x + 16) = \frac{4 \cdot - 4 + 4 (x + 4)}{4}$

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

$f (4 x + 16) = \frac{4 (- 4 + x + 4)}{4}$

Cancel the common factors.

Factor $4$ out of $4$ .

$f (4 x + 16) = \frac{4 (- 4 + x + 4)}{4 (1)}$

Cancel the common factor.

$f (4 x + 16) = \frac{4 (- 4 + x + 4)}{4 \cdot 1}$

Rewrite the expression.

$f (4 x + 16) = \frac{- 4 + x + 4}{1}$

Divide $- 4 + x + 4$ by $1$ .

$f (4 x + 16) = - 4 + x + 4$

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

Add $- 4$ and $4$ .

$f (4 x + 16) = x + 0$

Add $x$ and $0$ .

$f (4 x + 16) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 4 x + 16$ is the inverse of $f (x) = \frac{- 16 + x}{4}$ .

$f^{- 1} (x) = 4 x + 16$

Find the Inverse f(x)=(-16+x)/4

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