Find the Inverse 4c+5

$4 c + 5$

Step 1

Interchange the variables.

$c = 4 y + 5$

Step 2

Solve for $y$ .

Rewrite the equation as $4 y + 5 = c$ .

$4 y + 5 = c$

Subtract $5$ from both sides of the equation.

$4 y = c - 5$

Divide each term in $4 y = c - 5$ by $4$ and simplify.

Divide each term in $4 y = c - 5$ by $4$ .

$\frac{4 y}{4} = \frac{c}{4} + \frac{- 5}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{c}{4} + \frac{- 5}{4}$

Divide $y$ by $1$ .

$y = \frac{c}{4} + \frac{- 5}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{c}{4} - \frac{5}{4}$

Step 3

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

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

Step 4

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

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

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

Set up the composite result function.

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

Evaluate $f^{- 1} (4 c + 5)$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Combine the numerators over the common denominator.

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

Combine the opposite terms in $4 c + 5 - 5$ .

Subtract $5$ from $5$ .

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

Add $4 c$ and $0$ .

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $c$ by $1$ .

$f^{- 1} (4 c + 5) = c$

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

Set up the composite result function.

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

Evaluate $f (\frac{c}{4} - \frac{5}{4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{c}{4} - \frac{5}{4}) = 4 (\frac{c}{4} - \frac{5}{4}) + 5$

Simplify each term.

Apply the distributive property.

$f (\frac{c}{4} - \frac{5}{4}) = 4 (\frac{c}{4}) + 4 (- \frac{5}{4}) + 5$

Cancel the common factor of $4$ .

Cancel the common factor.

$f (\frac{c}{4} - \frac{5}{4}) = 4 (\frac{c}{4}) + 4 (- \frac{5}{4}) + 5$

Rewrite the expression.

$f (\frac{c}{4} - \frac{5}{4}) = c + 4 (- \frac{5}{4}) + 5$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$f (\frac{c}{4} - \frac{5}{4}) = c + 4 (\frac{- 5}{4}) + 5$

Cancel the common factor.

$f (\frac{c}{4} - \frac{5}{4}) = c + 4 (\frac{- 5}{4}) + 5$

Rewrite the expression.

$f (\frac{c}{4} - \frac{5}{4}) = c - 5 + 5$

Combine the opposite terms in $c - 5 + 5$ .

Add $- 5$ and $5$ .

$f (\frac{c}{4} - \frac{5}{4}) = c + 0$

Add $c$ and $0$ .

$f (\frac{c}{4} - \frac{5}{4}) = c$

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

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

Find the Inverse 4c+5

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