Find the Inverse 100000-2500x

$100000 - 2500 x$

Step 1

Interchange the variables.

$x = 100000 - 2500 y$

Step 2

Solve for $y$ .

Rewrite the equation as $100000 - 2500 y = x$ .

$100000 - 2500 y = x$

Subtract $100000$ from both sides of the equation.

$- 2500 y = x - 100000$

Divide each term in $- 2500 y = x - 100000$ by $- 2500$ and simplify.

Divide each term in $- 2500 y = x - 100000$ by $- 2500$ .

$\frac{- 2500 y}{- 2500} = \frac{x}{- 2500} + \frac{- 100000}{- 2500}$

Simplify the left side.

Cancel the common factor of $- 2500$ .

Cancel the common factor.

$\frac{- 2500 y}{- 2500} = \frac{x}{- 2500} + \frac{- 100000}{- 2500}$

Divide $y$ by $1$ .

$y = \frac{x}{- 2500} + \frac{- 100000}{- 2500}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{x}{2500} + \frac{- 100000}{- 2500}$

Divide $- 100000$ by $- 2500$ .

$y = - \frac{x}{2500} + 40$

Step 3

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

$f^{- 1} (x) = - \frac{x}{2500} + 40$

Step 4

Verify if $f^{- 1} (x) = - \frac{x}{2500} + 40$ is the inverse of $f (x) = 100000 - 2500 x$ .

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 $f^{- 1} (100000 - 2500 x)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (100000 - 2500 x) = - \frac{100000 - 2500 x}{2500} + 40$

Simplify each term.

Cancel the common factor of $100000 - 2500 x$ and $2500$ .

Factor $2500$ out of $100000$ .

$f^{- 1} (100000 - 2500 x) = - \frac{2500 \cdot 40 - 2500 x}{2500} + 40$

Factor $2500$ out of $- 2500 x$ .

$f^{- 1} (100000 - 2500 x) = - \frac{2500 \cdot 40 + 2500 (- x)}{2500} + 40$

Factor $2500$ out of $2500 (40) + 2500 (- x)$ .

$f^{- 1} (100000 - 2500 x) = - \frac{2500 (40 - x)}{2500} + 40$

Cancel the common factors.

Factor $2500$ out of $2500$ .

$f^{- 1} (100000 - 2500 x) = - \frac{2500 (40 - x)}{2500 (1)} + 40$

Cancel the common factor.

$f^{- 1} (100000 - 2500 x) = - \frac{2500 (40 - x)}{2500 \cdot 1} + 40$

Rewrite the expression.

$f^{- 1} (100000 - 2500 x) = - \frac{40 - x}{1} + 40$

Divide $40 - x$ by $1$ .

$f^{- 1} (100000 - 2500 x) = - (40 - x) + 40$

Apply the distributive property.

$f^{- 1} (100000 - 2500 x) = - 1 \cdot 40 + x + 40$

Multiply $- 1$ by $40$ .

$f^{- 1} (100000 - 2500 x) = - 40 + x + 40$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$f^{- 1} (100000 - 2500 x) = - 40 + 1 x + 40$

Multiply $x$ by $1$ .

$f^{- 1} (100000 - 2500 x) = - 40 + x + 40$

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

Add $- 40$ and $40$ .

$f^{- 1} (100000 - 2500 x) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (100000 - 2500 x) = x$

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

Set up the composite result function.

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

Evaluate $f (- \frac{x}{2500} + 40)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{x}{2500} + 40) = 100000 - 2500 (- \frac{x}{2500} + 40)$

Simplify each term.

Apply the distributive property.

$f (- \frac{x}{2500} + 40) = 100000 - 2500 (- \frac{x}{2500}) - 2500 \cdot 40$

Cancel the common factor of $2500$ .

Move the leading negative in $- \frac{x}{2500}$ into the numerator.

$f (- \frac{x}{2500} + 40) = 100000 - 2500 \frac{- x}{2500} - 2500 \cdot 40$

Factor $2500$ out of $- 2500$ .

$f (- \frac{x}{2500} + 40) = 100000 + 2500 (- 1) (\frac{- x}{2500}) - 2500 \cdot 40$

Cancel the common factor.

$f (- \frac{x}{2500} + 40) = 100000 + 2500 \cdot (- 1 \frac{- x}{2500}) - 2500 \cdot 40$

Rewrite the expression.

$f (- \frac{x}{2500} + 40) = 100000 - 1 (- x) - 2500 \cdot 40$

Multiply $- 1$ by $- 1$ .

$f (- \frac{x}{2500} + 40) = 100000 + 1 x - 2500 \cdot 40$

Multiply $x$ by $1$ .

$f (- \frac{x}{2500} + 40) = 100000 + x - 2500 \cdot 40$

Multiply $- 2500$ by $40$ .

$f (- \frac{x}{2500} + 40) = 100000 + x - 100000$

Combine the opposite terms in $100000 + x - 100000$ .

Subtract $100000$ from $100000$ .

$f (- \frac{x}{2500} + 40) = x + 0$

Add $x$ and $0$ .

$f (- \frac{x}{2500} + 40) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{x}{2500} + 40$ is the inverse of $f (x) = 100000 - 2500 x$ .

$f^{- 1} (x) = - \frac{x}{2500} + 40$

Find the Inverse 100000-2500x

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