Find the Inverse f(x)=4x+6

$f (x) = 4 x + 6$

Write $f (x) = 4 x + 6$ as an equation.

$y = 4 x + 6$

Interchange the variables.

$x = 4 y + 6$

Solve for $y$ .

Rewrite the equation as $4 y + 6 = x$ .

$4 y + 6 = x$

Subtract $6$ from both sides of the equation.

$4 y = x - 6$

Divide each term in $4 y = x - 6$ by $4$ and simplify.

Divide each term in $4 y = x - 6$ by $4$ .

$\frac{4 y}{4} = \frac{x}{4} + \frac{- 6}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{x}{4} + \frac{- 6}{4}$

Divide $y$ by $1$ .

$y = \frac{x}{4} + \frac{- 6}{4}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 6$ and $4$ .

Factor $2$ out of $- 6$ .

$y = \frac{x}{4} + \frac{2 (- 3)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{x}{4} + \frac{2 \cdot - 3}{2 \cdot 2}$

Cancel the common factor.

$y = \frac{x}{4} + \frac{2 \cdot - 3}{2 \cdot 2}$

Rewrite the expression.

$y = \frac{x}{4} + \frac{- 3}{2}$

Move the negative in front of the fraction.

$y = \frac{x}{4} - \frac{3}{2}$

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

$f^{- 1} (x) = \frac{x}{4} - \frac{3}{2}$

Verify if $f^{- 1} (x) = \frac{x}{4} - \frac{3}{2}$ is the inverse of $f (x) = 4 x + 6$ .

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 $f^{- 1} (4 x + 6)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (4 x + 6) = \frac{4 x + 6}{4} - \frac{3}{2}$

Cancel the common factor of $4 x + 6$ and $4$ .

Factor $2$ out of $4 x$ .

$f^{- 1} (4 x + 6) = \frac{2 (2 x) + 6}{4} - \frac{3}{2}$

Factor $2$ out of $6$ .

$f^{- 1} (4 x + 6) = \frac{2 (2 x) + 2 (3)}{4} - \frac{3}{2}$

Factor $2$ out of $2 (2 x) + 2 (3)$ .

$f^{- 1} (4 x + 6) = \frac{2 (2 x + 3)}{4} - \frac{3}{2}$

Cancel the common factors.

Factor $2$ out of $4$ .

$f^{- 1} (4 x + 6) = \frac{2 (2 x + 3)}{2 (2)} - \frac{3}{2}$

Cancel the common factor.

$f^{- 1} (4 x + 6) = \frac{2 (2 x + 3)}{2 \cdot 2} - \frac{3}{2}$

Rewrite the expression.

$f^{- 1} (4 x + 6) = \frac{2 x + 3}{2} - \frac{3}{2}$

Combine the numerators over the common denominator.

$f^{- 1} (4 x + 6) = \frac{2 x + 3 - 3}{2}$

Combine the opposite terms in $2 x + 3 - 3$ .

Subtract $3$ from $3$ .

$f^{- 1} (4 x + 6) = \frac{2 x + 0}{2}$

Add $2 x$ and $0$ .

$f^{- 1} (4 x + 6) = \frac{2 x}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (4 x + 6) = \frac{2 x}{2}$

Divide $x$ by $1$ .

$f^{- 1} (4 x + 6) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x}{4} - \frac{3}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{4} - \frac{3}{2}) = 4 (\frac{x}{4} - \frac{3}{2}) + 6$

Simplify each term.

Apply the distributive property.

$f (\frac{x}{4} - \frac{3}{2}) = 4 (\frac{x}{4}) + 4 (- \frac{3}{2}) + 6$

Cancel the common factor of $4$ .

Cancel the common factor.

$f (\frac{x}{4} - \frac{3}{2}) = 4 (\frac{x}{4}) + 4 (- \frac{3}{2}) + 6$

Rewrite the expression.

$f (\frac{x}{4} - \frac{3}{2}) = x + 4 (- \frac{3}{2}) + 6$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$f (\frac{x}{4} - \frac{3}{2}) = x + 4 (\frac{- 3}{2}) + 6$

Factor $2$ out of $4$ .

$f (\frac{x}{4} - \frac{3}{2}) = x + 2 (2) (\frac{- 3}{2}) + 6$

Cancel the common factor.

$f (\frac{x}{4} - \frac{3}{2}) = x + 2 \cdot (2 (\frac{- 3}{2})) + 6$

Rewrite the expression.

$f (\frac{x}{4} - \frac{3}{2}) = x + 2 \cdot - 3 + 6$

Multiply $2$ by $- 3$ .

$f (\frac{x}{4} - \frac{3}{2}) = x - 6 + 6$

Combine the opposite terms in $x - 6 + 6$ .

Add $- 6$ and $6$ .

$f (\frac{x}{4} - \frac{3}{2}) = x + 0$

Add $x$ and $0$ .

$f (\frac{x}{4} - \frac{3}{2}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{4} - \frac{3}{2}$ is the inverse of $f (x) = 4 x + 6$ .

$f^{- 1} (x) = \frac{x}{4} - \frac{3}{2}$

Find the Inverse f(x)=4x+6

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