Find the Inverse x/4+10

$\frac{x}{4} + 10$

Step 1

Interchange the variables.

$x = \frac{y}{4} + 10$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{y}{4} + 10 = x$ .

$\frac{y}{4} + 10 = x$

Subtract $10$ from both sides of the equation.

$\frac{y}{4} = x - 10$

Multiply both sides of the equation by $4$ .

$4 \frac{y}{4} = 4 (x - 10)$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$4 \frac{y}{4} = 4 (x - 10)$

Rewrite the expression.

$y = 4 (x - 10)$

Simplify the right side.

Simplify $4 (x - 10)$ .

Apply the distributive property.

$y = 4 x + 4 \cdot - 10$

Multiply $4$ by $- 10$ .

$y = 4 x - 40$

Step 3

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

$f^{- 1} (x) = 4 x - 40$

Step 4

Verify if $f^{- 1} (x) = 4 x - 40$ is the inverse of $f (x) = \frac{x}{4} + 10$ .

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 $f^{- 1} (\frac{x}{4} + 10)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x}{4} + 10) = 4 (\frac{x}{4} + 10) - 40$

Simplify each term.

Apply the distributive property.

$f^{- 1} (\frac{x}{4} + 10) = 4 (\frac{x}{4}) + 4 \cdot 10 - 40$

Cancel the common factor of $4$ .

Cancel the common factor.

$f^{- 1} (\frac{x}{4} + 10) = 4 (\frac{x}{4}) + 4 \cdot 10 - 40$

Rewrite the expression.

$f^{- 1} (\frac{x}{4} + 10) = x + 4 \cdot 10 - 40$

Multiply $4$ by $10$ .

$f^{- 1} (\frac{x}{4} + 10) = x + 40 - 40$

Combine the opposite terms in $x + 40 - 40$ .

Subtract $40$ from $40$ .

$f^{- 1} (\frac{x}{4} + 10) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{x}{4} + 10) = x$

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

Set up the composite result function.

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

Evaluate $f (4 x - 40)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (4 x - 40) = \frac{4 x - 40}{4} + 10$

Cancel the common factor of $4 x - 40$ and $4$ .

Factor $4$ out of $4 x$ .

$f (4 x - 40) = \frac{4 (x) - 40}{4} + 10$

Factor $4$ out of $- 40$ .

$f (4 x - 40) = \frac{4 (x) + 4 \cdot - 10}{4} + 10$

Factor $4$ out of $4 (x) + 4 (- 10)$ .

$f (4 x - 40) = \frac{4 (x - 10)}{4} + 10$

Cancel the common factors.

Factor $4$ out of $4$ .

$f (4 x - 40) = \frac{4 (x - 10)}{4 (1)} + 10$

Cancel the common factor.

$f (4 x - 40) = \frac{4 (x - 10)}{4 \cdot 1} + 10$

Rewrite the expression.

$f (4 x - 40) = \frac{x - 10}{1} + 10$

Divide $x - 10$ by $1$ .

$f (4 x - 40) = x - 10 + 10$

Combine the opposite terms in $x - 10 + 10$ .

Add $- 10$ and $10$ .

$f (4 x - 40) = x + 0$

Add $x$ and $0$ .

$f (4 x - 40) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 4 x - 40$ is the inverse of $f (x) = \frac{x}{4} + 10$ .

$f^{- 1} (x) = 4 x - 40$

Find the Inverse x/4+10

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