Find the Inverse 5x-200

$5 x - 200$

Step 1

Interchange the variables.

$x = 5 y - 200$

Step 2

Solve for $y$ .

Rewrite the equation as $5 y - 200 = x$ .

$5 y - 200 = x$

Add $200$ to both sides of the equation.

$5 y = x + 200$

Divide each term in $5 y = x + 200$ by $5$ and simplify.

Divide each term in $5 y = x + 200$ by $5$ .

$\frac{5 y}{5} = \frac{x}{5} + \frac{200}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 y}{5} = \frac{x}{5} + \frac{200}{5}$

Divide $y$ by $1$ .

$y = \frac{x}{5} + \frac{200}{5}$

Simplify the right side.

Divide $200$ by $5$ .

$y = \frac{x}{5} + 40$

Step 3

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

$f^{- 1} (x) = \frac{x}{5} + 40$

Step 4

Verify if $f^{- 1} (x) = \frac{x}{5} + 40$ is the inverse of $f (x) = 5 x - 200$ .

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 $f^{- 1} (5 x - 200)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (5 x - 200) = \frac{5 x - 200}{5} + 40$

Cancel the common factor of $5 x - 200$ and $5$ .

Factor $5$ out of $5 x$ .

$f^{- 1} (5 x - 200) = \frac{5 (x) - 200}{5} + 40$

Factor $5$ out of $- 200$ .

$f^{- 1} (5 x - 200) = \frac{5 (x) + 5 \cdot - 40}{5} + 40$

Factor $5$ out of $5 (x) + 5 (- 40)$ .

$f^{- 1} (5 x - 200) = \frac{5 (x - 40)}{5} + 40$

Cancel the common factors.

Factor $5$ out of $5$ .

$f^{- 1} (5 x - 200) = \frac{5 (x - 40)}{5 (1)} + 40$

Cancel the common factor.

$f^{- 1} (5 x - 200) = \frac{5 (x - 40)}{5 \cdot 1} + 40$

Rewrite the expression.

$f^{- 1} (5 x - 200) = \frac{x - 40}{1} + 40$

Divide $x - 40$ by $1$ .

$f^{- 1} (5 x - 200) = x - 40 + 40$

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

Add $- 40$ and $40$ .

$f^{- 1} (5 x - 200) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (5 x - 200) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x}{5} + 40)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{5} + 40) = 5 (\frac{x}{5} + 40) - 200$

Simplify each term.

Apply the distributive property.

$f (\frac{x}{5} + 40) = 5 (\frac{x}{5}) + 5 \cdot 40 - 200$

Cancel the common factor of $5$ .

Cancel the common factor.

$f (\frac{x}{5} + 40) = 5 (\frac{x}{5}) + 5 \cdot 40 - 200$

Rewrite the expression.

$f (\frac{x}{5} + 40) = x + 5 \cdot 40 - 200$

Multiply $5$ by $40$ .

$f (\frac{x}{5} + 40) = x + 200 - 200$

Combine the opposite terms in $x + 200 - 200$ .

Subtract $200$ from $200$ .

$f (\frac{x}{5} + 40) = x + 0$

Add $x$ and $0$ .

$f (\frac{x}{5} + 40) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{5} + 40$ is the inverse of $f (x) = 5 x - 200$ .

$f^{- 1} (x) = \frac{x}{5} + 40$

Find the Inverse 5x-200

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