Find the Inverse f(x)=9x+3

$f (x) = 9 x + 3$

Step 1

Write $f (x) = 9 x + 3$ as an equation.

$y = 9 x + 3$

Step 2

Interchange the variables.

$x = 9 y + 3$

Step 3

Solve for $y$ .

Rewrite the equation as $9 y + 3 = x$ .

$9 y + 3 = x$

Subtract $3$ from both sides of the equation.

$9 y = x - 3$

Divide each term in $9 y = x - 3$ by $9$ and simplify.

Divide each term in $9 y = x - 3$ by $9$ .

$\frac{9 y}{9} = \frac{x}{9} + \frac{- 3}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 y}{9} = \frac{x}{9} + \frac{- 3}{9}$

Divide $y$ by $1$ .

$y = \frac{x}{9} + \frac{- 3}{9}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 3$ and $9$ .

Factor $3$ out of $- 3$ .

$y = \frac{x}{9} + \frac{3 (- 1)}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

$y = \frac{x}{9} + \frac{3 \cdot - 1}{3 \cdot 3}$

Cancel the common factor.

$y = \frac{x}{9} + \frac{3 \cdot - 1}{3 \cdot 3}$

Rewrite the expression.

$y = \frac{x}{9} + \frac{- 1}{3}$

Move the negative in front of the fraction.

$y = \frac{x}{9} - \frac{1}{3}$

Step 4

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

$f^{- 1} (x) = \frac{x}{9} - \frac{1}{3}$

Step 5

Verify if $f^{- 1} (x) = \frac{x}{9} - \frac{1}{3}$ is the inverse of $f (x) = 9 x + 3$ .

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 $f^{- 1} (9 x + 3)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (9 x + 3) = \frac{9 x + 3}{9} - \frac{1}{3}$

Cancel the common factor of $9 x + 3$ and $9$ .

Factor $3$ out of $9 x$ .

$f^{- 1} (9 x + 3) = \frac{3 (3 x) + 3}{9} - \frac{1}{3}$

Factor $3$ out of $3$ .

$f^{- 1} (9 x + 3) = \frac{3 (3 x) + 3 (1)}{9} - \frac{1}{3}$

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

$f^{- 1} (9 x + 3) = \frac{3 (3 x + 1)}{9} - \frac{1}{3}$

Cancel the common factors.

Factor $3$ out of $9$ .

$f^{- 1} (9 x + 3) = \frac{3 (3 x + 1)}{3 (3)} - \frac{1}{3}$

Cancel the common factor.

$f^{- 1} (9 x + 3) = \frac{3 (3 x + 1)}{3 \cdot 3} - \frac{1}{3}$

Rewrite the expression.

$f^{- 1} (9 x + 3) = \frac{3 x + 1}{3} - \frac{1}{3}$

Combine the numerators over the common denominator.

$f^{- 1} (9 x + 3) = \frac{3 x + 1 - 1}{3}$

Combine the opposite terms in $3 x + 1 - 1$ .

Subtract $1$ from $1$ .

$f^{- 1} (9 x + 3) = \frac{3 x + 0}{3}$

Add $3 x$ and $0$ .

$f^{- 1} (9 x + 3) = \frac{3 x}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (9 x + 3) = \frac{3 x}{3}$

Divide $x$ by $1$ .

$f^{- 1} (9 x + 3) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x}{9} - \frac{1}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{9} - \frac{1}{3}) = 9 (\frac{x}{9} - \frac{1}{3}) + 3$

Simplify each term.

Apply the distributive property.

$f (\frac{x}{9} - \frac{1}{3}) = 9 (\frac{x}{9}) + 9 (- \frac{1}{3}) + 3$

Cancel the common factor of $9$ .

Cancel the common factor.

$f (\frac{x}{9} - \frac{1}{3}) = 9 (\frac{x}{9}) + 9 (- \frac{1}{3}) + 3$

Rewrite the expression.

$f (\frac{x}{9} - \frac{1}{3}) = x + 9 (- \frac{1}{3}) + 3$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$f (\frac{x}{9} - \frac{1}{3}) = x + 9 (\frac{- 1}{3}) + 3$

Factor $3$ out of $9$ .

$f (\frac{x}{9} - \frac{1}{3}) = x + 3 (3) (\frac{- 1}{3}) + 3$

Cancel the common factor.

$f (\frac{x}{9} - \frac{1}{3}) = x + 3 \cdot (3 (\frac{- 1}{3})) + 3$

Rewrite the expression.

$f (\frac{x}{9} - \frac{1}{3}) = x + 3 \cdot - 1 + 3$

Multiply $3$ by $- 1$ .

$f (\frac{x}{9} - \frac{1}{3}) = x - 3 + 3$

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

Add $- 3$ and $3$ .

$f (\frac{x}{9} - \frac{1}{3}) = x + 0$

Add $x$ and $0$ .

$f (\frac{x}{9} - \frac{1}{3}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{9} - \frac{1}{3}$ is the inverse of $f (x) = 9 x + 3$ .

$f^{- 1} (x) = \frac{x}{9} - \frac{1}{3}$

Find the Inverse f(x)=9x+3

Related Questions

Support LCMGCF.com by disabling your adblocker.