Find the Inverse f(x)=6x+3

$f (x) = 6 x + 3$

Write $f (x) = 6 x + 3$ as an equation.

$y = 6 x + 3$

Interchange the variables.

$x = 6 y + 3$

Solve for $y$ .

Rewrite the equation as $6 y + 3 = x$ .

$6 y + 3 = x$

Subtract $3$ from both sides of the equation.

$6 y = x - 3$

Divide each term in $6 y = x - 3$ by $6$ and simplify.

Divide each term in $6 y = x - 3$ by $6$ .

$\frac{6 y}{6} = \frac{x}{6} + \frac{- 3}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} = \frac{x}{6} + \frac{- 3}{6}$

Divide $y$ by $1$ .

$y = \frac{x}{6} + \frac{- 3}{6}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 3$ and $6$ .

Factor $3$ out of $- 3$ .

$y = \frac{x}{6} + \frac{3 (- 1)}{6}$

Cancel the common factors.

Factor $3$ out of $6$ .

$y = \frac{x}{6} + \frac{3 \cdot - 1}{3 \cdot 2}$

Cancel the common factor.

$y = \frac{x}{6} + \frac{3 \cdot - 1}{3 \cdot 2}$

Rewrite the expression.

$y = \frac{x}{6} + \frac{- 1}{2}$

Move the negative in front of the fraction.

$y = \frac{x}{6} - \frac{1}{2}$

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

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

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

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 $f^{- 1} (6 x + 3)$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Cancel the common factor of $6 x + 3$ and $6$ .

Factor $3$ out of $6 x$ .

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

Factor $3$ out of $3$ .

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

Factor $3$ out of $3 (2 x) + 3 (1)$ .

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

Cancel the common factors.

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Combine the numerators over the common denominator.

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

Combine the opposite terms in $2 x + 1 - 1$ .

Subtract $1$ from $1$ .

$f^{- 1} (6 x + 3) = \frac{2 x + 0}{2}$

Add $2 x$ and $0$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

$f^{- 1} (6 x + 3) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x}{6} - \frac{1}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $6$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $- 1$ .

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

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

Add $- 3$ and $3$ .

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

Add $x$ and $0$ .

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

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

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

Find the Inverse f(x)=6x+3

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