Find the Inverse y=ax+b

$y = a x + b$

Interchange the variables.

$a = y x + b$

Solve for $y$ .

Rewrite the equation as $y x + b = a$ .

$y x + b = a$

Subtract $b$ from both sides of the equation.

$y x = a - b$

Divide each term in $y x = a - b$ by $x$ and simplify.

Divide each term in $y x = a - b$ by $x$ .

$\frac{y x}{x} = \frac{a}{x} + \frac{- b}{x}$

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{y x}{x} = \frac{a}{x} + \frac{- b}{x}$

Divide $y$ by $1$ .

$y = \frac{a}{x} + \frac{- b}{x}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{a}{x} - \frac{b}{x}$

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

$f^{- 1} (a) = \frac{a}{x} - \frac{b}{x}$

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

To verify the inverse, check if $f^{- 1} (f (a)) = a$ and $f (f^{- 1} (a)) = a$ .

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

Set up the composite result function.

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

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

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

Combine the numerators over the common denominator.

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

Combine the opposite terms in $a x + b - b$ .

Subtract $b$ from $b$ .

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

Add $a x$ and $0$ .

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Divide $a$ by $1$ .

$f^{- 1} (a x + b) = a$

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

Set up the composite result function.

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

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

$f (\frac{a}{x} - \frac{b}{x}) = (\frac{a}{x} - \frac{b}{x}) \cdot x + b$

Simplify each term.

Apply the distributive property.

$f (\frac{a}{x} - \frac{b}{x}) = \frac{a}{x} \cdot x - \frac{b}{x} \cdot x + b$

Cancel the common factor of $x$ .

Cancel the common factor.

$f (\frac{a}{x} - \frac{b}{x}) = \frac{a}{x} \cdot x - \frac{b}{x} \cdot x + b$

Rewrite the expression.

$f (\frac{a}{x} - \frac{b}{x}) = a - \frac{b}{x} \cdot x + b$

Cancel the common factor of $x$ .

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

$f (\frac{a}{x} - \frac{b}{x}) = a + \frac{- b}{x} \cdot x + b$

Cancel the common factor.

$f (\frac{a}{x} - \frac{b}{x}) = a + \frac{- b}{x} \cdot x + b$

Rewrite the expression.

$f (\frac{a}{x} - \frac{b}{x}) = a - b + b$

Combine the opposite terms in $a - b + b$ .

Add $- b$ and $b$ .

$f (\frac{a}{x} - \frac{b}{x}) = a + 0$

Add $a$ and $0$ .

$f (\frac{a}{x} - \frac{b}{x}) = a$

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

$f^{- 1} (a) = \frac{a}{x} - \frac{b}{x}$

Find the Inverse y=ax+b

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