Find the Inverse 12x-3

$12 x - 3$

Step 1

Interchange the variables.

$x = 12 y - 3$

Step 2

Solve for $y$ .

Rewrite the equation as $12 y - 3 = x$ .

$12 y - 3 = x$

Add $3$ to both sides of the equation.

$12 y = x + 3$

Divide each term in $12 y = x + 3$ by $12$ and simplify.

Divide each term in $12 y = x + 3$ by $12$ .

$\frac{12 y}{12} = \frac{x}{12} + \frac{3}{12}$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 y}{12} = \frac{x}{12} + \frac{3}{12}$

Divide $y$ by $1$ .

$y = \frac{x}{12} + \frac{3}{12}$

Simplify the right side.

Cancel the common factor of $3$ and $12$ .

Factor $3$ out of $3$ .

$y = \frac{x}{12} + \frac{3 (1)}{12}$

Cancel the common factors.

Factor $3$ out of $12$ .

$y = \frac{x}{12} + \frac{3 \cdot 1}{3 \cdot 4}$

Cancel the common factor.

$y = \frac{x}{12} + \frac{3 \cdot 1}{3 \cdot 4}$

Rewrite the expression.

$y = \frac{x}{12} + \frac{1}{4}$

Step 3

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

$f^{- 1} (x) = \frac{x}{12} + \frac{1}{4}$

Step 4

Verify if $f^{- 1} (x) = \frac{x}{12} + \frac{1}{4}$ is the inverse of $f (x) = 12 x - 3$ .

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 $f^{- 1} (12 x - 3)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (12 x - 3) = \frac{12 x - 3}{12} + \frac{1}{4}$

Cancel the common factor of $12 x - 3$ and $12$ .

Factor $3$ out of $12 x$ .

$f^{- 1} (12 x - 3) = \frac{3 (4 x) - 3}{12} + \frac{1}{4}$

Factor $3$ out of $- 3$ .

$f^{- 1} (12 x - 3) = \frac{3 (4 x) + 3 (- 1)}{12} + \frac{1}{4}$

Factor $3$ out of $3 (4 x) + 3 (- 1)$ .

$f^{- 1} (12 x - 3) = \frac{3 (4 x - 1)}{12} + \frac{1}{4}$

Cancel the common factors.

Factor $3$ out of $12$ .

$f^{- 1} (12 x - 3) = \frac{3 (4 x - 1)}{3 (4)} + \frac{1}{4}$

Cancel the common factor.

$f^{- 1} (12 x - 3) = \frac{3 (4 x - 1)}{3 \cdot 4} + \frac{1}{4}$

Rewrite the expression.

$f^{- 1} (12 x - 3) = \frac{4 x - 1}{4} + \frac{1}{4}$

Combine the numerators over the common denominator.

$f^{- 1} (12 x - 3) = \frac{4 x - 1 + 1}{4}$

Combine the opposite terms in $4 x - 1 + 1$ .

Add $- 1$ and $1$ .

$f^{- 1} (12 x - 3) = \frac{4 x + 0}{4}$

Add $4 x$ and $0$ .

$f^{- 1} (12 x - 3) = \frac{4 x}{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$f^{- 1} (12 x - 3) = \frac{4 x}{4}$

Divide $x$ by $1$ .

$f^{- 1} (12 x - 3) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x}{12} + \frac{1}{4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{12} + \frac{1}{4}) = 12 (\frac{x}{12} + \frac{1}{4}) - 3$

Simplify each term.

Apply the distributive property.

$f (\frac{x}{12} + \frac{1}{4}) = 12 (\frac{x}{12}) + 12 (\frac{1}{4}) - 3$

Cancel the common factor of $12$ .

Cancel the common factor.

$f (\frac{x}{12} + \frac{1}{4}) = 12 (\frac{x}{12}) + 12 (\frac{1}{4}) - 3$

Rewrite the expression.

$f (\frac{x}{12} + \frac{1}{4}) = x + 12 (\frac{1}{4}) - 3$

Cancel the common factor of $4$ .

Factor $4$ out of $12$ .

$f (\frac{x}{12} + \frac{1}{4}) = x + 4 (3) (\frac{1}{4}) - 3$

Cancel the common factor.

$f (\frac{x}{12} + \frac{1}{4}) = x + 4 \cdot (3 (\frac{1}{4})) - 3$

Rewrite the expression.

$f (\frac{x}{12} + \frac{1}{4}) = x + 3 - 3$

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

Subtract $3$ from $3$ .

$f (\frac{x}{12} + \frac{1}{4}) = x + 0$

Add $x$ and $0$ .

$f (\frac{x}{12} + \frac{1}{4}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{12} + \frac{1}{4}$ is the inverse of $f (x) = 12 x - 3$ .

$f^{- 1} (x) = \frac{x}{12} + \frac{1}{4}$

Find the Inverse 12x-3

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