Find the Inverse x/5+1

$\frac{x}{5} + 1$

Step 1

Interchange the variables.

$x = \frac{y}{5} + 1$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{y}{5} + 1 = x$ .

$\frac{y}{5} + 1 = x$

Subtract $1$ from both sides of the equation.

$\frac{y}{5} = x - 1$

Multiply both sides of the equation by $5$ .

$5 \frac{y}{5} = 5 (x - 1)$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$5 \frac{y}{5} = 5 (x - 1)$

Rewrite the expression.

$y = 5 (x - 1)$

Simplify the right side.

Simplify $5 (x - 1)$ .

Apply the distributive property.

$y = 5 x + 5 \cdot - 1$

Multiply $5$ by $- 1$ .

$y = 5 x - 5$

Step 3

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

$f^{- 1} (x) = 5 x - 5$

Step 4

Verify if $f^{- 1} (x) = 5 x - 5$ is the inverse of $f (x) = \frac{x}{5} + 1$ .

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 $f^{- 1} (\frac{x}{5} + 1)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x}{5} + 1) = 5 (\frac{x}{5} + 1) - 5$

Simplify each term.

Apply the distributive property.

$f^{- 1} (\frac{x}{5} + 1) = 5 (\frac{x}{5}) + 5 \cdot 1 - 5$

Cancel the common factor of $5$ .

Cancel the common factor.

$f^{- 1} (\frac{x}{5} + 1) = 5 (\frac{x}{5}) + 5 \cdot 1 - 5$

Rewrite the expression.

$f^{- 1} (\frac{x}{5} + 1) = x + 5 \cdot 1 - 5$

Multiply $5$ by $1$ .

$f^{- 1} (\frac{x}{5} + 1) = x + 5 - 5$

Combine the opposite terms in $x + 5 - 5$ .

Subtract $5$ from $5$ .

$f^{- 1} (\frac{x}{5} + 1) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{x}{5} + 1) = x$

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

Set up the composite result function.

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

Evaluate $f (5 x - 5)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (5 x - 5) = \frac{5 x - 5}{5} + 1$

Cancel the common factor of $5 x - 5$ and $5$ .

Factor $5$ out of $5 x$ .

$f (5 x - 5) = \frac{5 (x) - 5}{5} + 1$

Factor $5$ out of $- 5$ .

$f (5 x - 5) = \frac{5 (x) + 5 \cdot - 1}{5} + 1$

Factor $5$ out of $5 (x) + 5 (- 1)$ .

$f (5 x - 5) = \frac{5 (x - 1)}{5} + 1$

Cancel the common factors.

Factor $5$ out of $5$ .

$f (5 x - 5) = \frac{5 (x - 1)}{5 (1)} + 1$

Cancel the common factor.

$f (5 x - 5) = \frac{5 (x - 1)}{5 \cdot 1} + 1$

Rewrite the expression.

$f (5 x - 5) = \frac{x - 1}{1} + 1$

Divide $x - 1$ by $1$ .

$f (5 x - 5) = x - 1 + 1$

Combine the opposite terms in $x - 1 + 1$ .

Add $- 1$ and $1$ .

$f (5 x - 5) = x + 0$

Add $x$ and $0$ .

$f (5 x - 5) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 5 x - 5$ is the inverse of $f (x) = \frac{x}{5} + 1$ .

$f^{- 1} (x) = 5 x - 5$

Find the Inverse x/5+1

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