Find the Inverse f(x)=ax-b

$f (x) = a x - b$

Write $f (x) = a x - b$ as an equation.

$y = a x - b$

Interchange the variables.

$a = y x - b$

Solve for $y$ .

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

$y x - b = a$

Add $b$ to 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 $1$ .

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

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 $a x - b + b$ .

Add $- b$ and $b$ .

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

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 $x$ .

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 $a + b - b$ .

Subtract $b$ from $b$ .

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

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 f(x)=ax-b

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