Find the Inverse y=5x+20

$y = 5 x + 20$

Step 1

Interchange the variables.

$x = 5 y + 20$

Step 2

Solve for $y$ .

Rewrite the equation as $5 y + 20 = x$ .

$5 y + 20 = x$

Subtract $20$ from both sides of the equation.

$5 y = x - 20$

Divide each term in $5 y = x - 20$ by $5$ and simplify.

Divide each term in $5 y = x - 20$ by $5$ .

$\frac{5 y}{5} = \frac{x}{5} + \frac{- 20}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 y}{5} = \frac{x}{5} + \frac{- 20}{5}$

Divide $y$ by $1$ .

$y = \frac{x}{5} + \frac{- 20}{5}$

Simplify the right side.

Divide $- 20$ by $5$ .

$y = \frac{x}{5} - 4$

Step 3

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

$f^{- 1} (x) = \frac{x}{5} - 4$

Step 4

Verify if $f^{- 1} (x) = \frac{x}{5} - 4$ is the inverse of $f (x) = 5 x + 20$ .

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 $f^{- 1} (5 x + 20)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (5 x + 20) = \frac{5 x + 20}{5} - 4$

Cancel the common factor of $5 x + 20$ and $5$ .

Factor $5$ out of $5 x$ .

$f^{- 1} (5 x + 20) = \frac{5 (x) + 20}{5} - 4$

Factor $5$ out of $20$ .

$f^{- 1} (5 x + 20) = \frac{5 (x) + 5 \cdot 4}{5} - 4$

Factor $5$ out of $5 (x) + 5 (4)$ .

$f^{- 1} (5 x + 20) = \frac{5 (x + 4)}{5} - 4$

Cancel the common factors.

Factor $5$ out of $5$ .

$f^{- 1} (5 x + 20) = \frac{5 (x + 4)}{5 (1)} - 4$

Cancel the common factor.

$f^{- 1} (5 x + 20) = \frac{5 (x + 4)}{5 \cdot 1} - 4$

Rewrite the expression.

$f^{- 1} (5 x + 20) = \frac{x + 4}{1} - 4$

Divide $x + 4$ by $1$ .

$f^{- 1} (5 x + 20) = x + 4 - 4$

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

Subtract $4$ from $4$ .

$f^{- 1} (5 x + 20) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (5 x + 20) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x}{5} - 4)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{5} - 4) = 5 (\frac{x}{5} - 4) + 20$

Simplify each term.

Apply the distributive property.

$f (\frac{x}{5} - 4) = 5 (\frac{x}{5}) + 5 \cdot - 4 + 20$

Cancel the common factor of $5$ .

Cancel the common factor.

$f (\frac{x}{5} - 4) = 5 (\frac{x}{5}) + 5 \cdot - 4 + 20$

Rewrite the expression.

$f (\frac{x}{5} - 4) = x + 5 \cdot - 4 + 20$

Multiply $5$ by $- 4$ .

$f (\frac{x}{5} - 4) = x - 20 + 20$

Combine the opposite terms in $x - 20 + 20$ .

Add $- 20$ and $20$ .

$f (\frac{x}{5} - 4) = x + 0$

Add $x$ and $0$ .

$f (\frac{x}{5} - 4) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{5} - 4$ is the inverse of $f (x) = 5 x + 20$ .

$f^{- 1} (x) = \frac{x}{5} - 4$

Find the Inverse y=5x+20

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