Find the Inverse f(x)=x/5-2

$f (x) = \frac{x}{5} - 2$

Step 1

Write $f (x) = \frac{x}{5} - 2$ as an equation.

$y = \frac{x}{5} - 2$

Step 2

Interchange the variables.

$x = \frac{y}{5} - 2$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{y}{5} - 2 = x$ .

$\frac{y}{5} - 2 = x$

Add $2$ to both sides of the equation.

$\frac{y}{5} = x + 2$

Multiply both sides of the equation by $5$ .

$5 \frac{y}{5} = 5 (x + 2)$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$5 \frac{y}{5} = 5 (x + 2)$

Rewrite the expression.

$y = 5 (x + 2)$

Simplify the right side.

Simplify $5 (x + 2)$ .

Apply the distributive property.

$y = 5 x + 5 \cdot 2$

Multiply $5$ by $2$ .

$y = 5 x + 10$

Step 4

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

$f^{- 1} (x) = 5 x + 10$

Step 5

Verify if $f^{- 1} (x) = 5 x + 10$ is the inverse of $f (x) = \frac{x}{5} - 2$ .

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 $f^{- 1} (\frac{x}{5} - 2)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x}{5} - 2) = 5 (\frac{x}{5} - 2) + 10$

Simplify each term.

Apply the distributive property.

$f^{- 1} (\frac{x}{5} - 2) = 5 (\frac{x}{5}) + 5 \cdot - 2 + 10$

Cancel the common factor of $5$ .

Cancel the common factor.

$f^{- 1} (\frac{x}{5} - 2) = 5 (\frac{x}{5}) + 5 \cdot - 2 + 10$

Rewrite the expression.

$f^{- 1} (\frac{x}{5} - 2) = x + 5 \cdot - 2 + 10$

Multiply $5$ by $- 2$ .

$f^{- 1} (\frac{x}{5} - 2) = x - 10 + 10$

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

Add $- 10$ and $10$ .

$f^{- 1} (\frac{x}{5} - 2) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{x}{5} - 2) = x$

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

Set up the composite result function.

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

Evaluate $f (5 x + 10)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (5 x + 10) = \frac{5 x + 10}{5} - 2$

Cancel the common factor of $5 x + 10$ and $5$ .

Factor $5$ out of $5 x$ .

$f (5 x + 10) = \frac{5 (x) + 10}{5} - 2$

Factor $5$ out of $10$ .

$f (5 x + 10) = \frac{5 (x) + 5 \cdot 2}{5} - 2$

Factor $5$ out of $5 (x) + 5 (2)$ .

$f (5 x + 10) = \frac{5 (x + 2)}{5} - 2$

Cancel the common factors.

Factor $5$ out of $5$ .

$f (5 x + 10) = \frac{5 (x + 2)}{5 (1)} - 2$

Cancel the common factor.

$f (5 x + 10) = \frac{5 (x + 2)}{5 \cdot 1} - 2$

Rewrite the expression.

$f (5 x + 10) = \frac{x + 2}{1} - 2$

Divide $x + 2$ by $1$ .

$f (5 x + 10) = x + 2 - 2$

Combine the opposite terms in $x + 2 - 2$ .

Subtract $2$ from $2$ .

$f (5 x + 10) = x + 0$

Add $x$ and $0$ .

$f (5 x + 10) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 5 x + 10$ is the inverse of $f (x) = \frac{x}{5} - 2$ .

$f^{- 1} (x) = 5 x + 10$

Find the Inverse f(x)=x/5-2

Related Questions

Support LCMGCF.com by disabling your adblocker.