Find the Inverse (x+6)/4

$\frac{x + 6}{4}$

Step 1

Interchange the variables.

$x = \frac{y + 6}{4}$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{y + 6}{4} = x$ .

$\frac{y + 6}{4} = x$

Multiply both sides of the equation by $4$ .

$4 \frac{y + 6}{4} = 4 x$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$4 \frac{y + 6}{4} = 4 x$

Rewrite the expression.

$y + 6 = 4 x$

Subtract $6$ from both sides of the equation.

$y = 4 x - 6$

Step 3

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

$f^{- 1} (x) = 4 x - 6$

Step 4

Verify if $f^{- 1} (x) = 4 x - 6$ is the inverse of $f (x) = \frac{x + 6}{4}$ .

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 $f^{- 1} (\frac{x + 6}{4})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x + 6}{4}) = 4 (\frac{x + 6}{4}) - 6$

Cancel the common factor of $4$ .

Cancel the common factor.

$f^{- 1} (\frac{x + 6}{4}) = 4 (\frac{x + 6}{4}) - 6$

Rewrite the expression.

$f^{- 1} (\frac{x + 6}{4}) = x + 6 - 6$

Combine the opposite terms in $x + 6 - 6$ .

Subtract $6$ from $6$ .

$f^{- 1} (\frac{x + 6}{4}) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{x + 6}{4}) = x$

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

Set up the composite result function.

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

Evaluate $f (4 x - 6)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (4 x - 6) = \frac{(4 x - 6) + 6}{4}$

Cancel the common factor of $(4 x - 6) + 6$ and $4$ .

Factor $2$ out of $4 x$ .

$f (4 x - 6) = \frac{2 (2 x) - 6 + 6}{4}$

Factor $2$ out of $- 6$ .

$f (4 x - 6) = \frac{2 (2 x) + 2 (- 3) + 6}{4}$

Factor $2$ out of $2 (2 x) + 2 (- 3)$ .

$f (4 x - 6) = \frac{2 (2 x - 3) + 6}{4}$

Factor $2$ out of $6$ .

$f (4 x - 6) = \frac{2 (2 x - 3) + 2 (3)}{4}$

Factor $2$ out of $2 (2 x - 3) + 2 (3)$ .

$f (4 x - 6) = \frac{2 (2 x - 3 + 3)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$f (4 x - 6) = \frac{2 (2 x - 3 + 3)}{2 (2)}$

Cancel the common factor.

$f (4 x - 6) = \frac{2 (2 x - 3 + 3)}{2 \cdot 2}$

Rewrite the expression.

$f (4 x - 6) = \frac{2 x - 3 + 3}{2}$

Simplify the numerator.

Add $- 3$ and $3$ .

$f (4 x - 6) = \frac{2 x + 0}{2}$

Add $2 x$ and $0$ .

$f (4 x - 6) = \frac{2 x}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (4 x - 6) = \frac{2 x}{2}$

Divide $x$ by $1$ .

$f (4 x - 6) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 4 x - 6$ is the inverse of $f (x) = \frac{x + 6}{4}$ .

$f^{- 1} (x) = 4 x - 6$

Find the Inverse (x+6)/4

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