Find the Inverse 19x+4

$19 x + 4$

Step 1

Interchange the variables.

$x = 19 y + 4$

Step 2

Solve for $y$ .

Rewrite the equation as $19 y + 4 = x$ .

$19 y + 4 = x$

Subtract $4$ from both sides of the equation.

$19 y = x - 4$

Divide each term in $19 y = x - 4$ by $19$ and simplify.

Divide each term in $19 y = x - 4$ by $19$ .

$\frac{19 y}{19} = \frac{x}{19} + \frac{- 4}{19}$

Simplify the left side.

Cancel the common factor of $19$ .

Cancel the common factor.

$\frac{19 y}{19} = \frac{x}{19} + \frac{- 4}{19}$

Divide $y$ by $1$ .

$y = \frac{x}{19} + \frac{- 4}{19}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{x}{19} - \frac{4}{19}$

Step 3

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

$f^{- 1} (x) = \frac{x}{19} - \frac{4}{19}$

Step 4

Verify if $f^{- 1} (x) = \frac{x}{19} - \frac{4}{19}$ is the inverse of $f (x) = 19 x + 4$ .

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 $f^{- 1} (19 x + 4)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (19 x + 4) = \frac{19 x + 4}{19} - \frac{4}{19}$

Combine the numerators over the common denominator.

$f^{- 1} (19 x + 4) = \frac{19 x + 4 - 4}{19}$

Combine the opposite terms in $19 x + 4 - 4$ .

Subtract $4$ from $4$ .

$f^{- 1} (19 x + 4) = \frac{19 x + 0}{19}$

Add $19 x$ and $0$ .

$f^{- 1} (19 x + 4) = \frac{19 x}{19}$

Cancel the common factor of $19$ .

Cancel the common factor.

$f^{- 1} (19 x + 4) = \frac{19 x}{19}$

Divide $x$ by $1$ .

$f^{- 1} (19 x + 4) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x}{19} - \frac{4}{19})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{19} - \frac{4}{19}) = 19 (\frac{x}{19} - \frac{4}{19}) + 4$

Simplify each term.

Apply the distributive property.

$f (\frac{x}{19} - \frac{4}{19}) = 19 (\frac{x}{19}) + 19 (- \frac{4}{19}) + 4$

Cancel the common factor of $19$ .

Cancel the common factor.

$f (\frac{x}{19} - \frac{4}{19}) = 19 (\frac{x}{19}) + 19 (- \frac{4}{19}) + 4$

Rewrite the expression.

$f (\frac{x}{19} - \frac{4}{19}) = x + 19 (- \frac{4}{19}) + 4$

Cancel the common factor of $19$ .

Move the leading negative in $- \frac{4}{19}$ into the numerator.

$f (\frac{x}{19} - \frac{4}{19}) = x + 19 (\frac{- 4}{19}) + 4$

Cancel the common factor.

$f (\frac{x}{19} - \frac{4}{19}) = x + 19 (\frac{- 4}{19}) + 4$

Rewrite the expression.

$f (\frac{x}{19} - \frac{4}{19}) = x - 4 + 4$

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

Add $- 4$ and $4$ .

$f (\frac{x}{19} - \frac{4}{19}) = x + 0$

Add $x$ and $0$ .

$f (\frac{x}{19} - \frac{4}{19}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{19} - \frac{4}{19}$ is the inverse of $f (x) = 19 x + 4$ .

$f^{- 1} (x) = \frac{x}{19} - \frac{4}{19}$

Find the Inverse 19x+4

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