Find the Inverse 20x-4

$20 x - 4$

Step 1

Interchange the variables.

$x = 20 y - 4$

Step 2

Solve for $y$ .

Rewrite the equation as $20 y - 4 = x$ .

$20 y - 4 = x$

Add $4$ to both sides of the equation.

$20 y = x + 4$

Divide each term in $20 y = x + 4$ by $20$ and simplify.

Divide each term in $20 y = x + 4$ by $20$ .

$\frac{20 y}{20} = \frac{x}{20} + \frac{4}{20}$

Simplify the left side.

Cancel the common factor of $20$ .

Cancel the common factor.

$\frac{20 y}{20} = \frac{x}{20} + \frac{4}{20}$

Divide $y$ by $1$ .

$y = \frac{x}{20} + \frac{4}{20}$

Simplify the right side.

Cancel the common factor of $4$ and $20$ .

Factor $4$ out of $4$ .

$y = \frac{x}{20} + \frac{4 (1)}{20}$

Cancel the common factors.

Factor $4$ out of $20$ .

$y = \frac{x}{20} + \frac{4 \cdot 1}{4 \cdot 5}$

Cancel the common factor.

$y = \frac{x}{20} + \frac{4 \cdot 1}{4 \cdot 5}$

Rewrite the expression.

$y = \frac{x}{20} + \frac{1}{5}$

Step 3

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

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

Step 4

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

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 $f^{- 1} (20 x - 4)$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Cancel the common factor of $20 x - 4$ and $20$ .

Factor $4$ out of $20 x$ .

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

Factor $4$ out of $- 4$ .

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

Factor $4$ out of $4 (5 x) + 4 (- 1)$ .

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

Cancel the common factors.

Factor $4$ out of $20$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine the numerators over the common denominator.

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

Combine the opposite terms in $5 x - 1 + 1$ .

Add $- 1$ and $1$ .

$f^{- 1} (20 x - 4) = \frac{5 x + 0}{5}$

Add $5 x$ and $0$ .

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $20$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $5$ .

Factor $5$ out of $20$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Subtract $4$ from $4$ .

$f (\frac{x}{20} + \frac{1}{5}) = x + 0$

Add $x$ and $0$ .

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

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

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

Find the Inverse 20x-4

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