Find the Inverse y=(x+3)/3

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{y + 3}{3} = x$ .

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

Multiply both sides of the equation by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$y + 3 = 3 x$

Subtract $3$ from both sides of the equation.

$y = 3 x - 3$

Step 3

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

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

Step 4

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Subtract $3$ from $3$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

Cancel the common factor of $(3 x - 3) + 3$ and $3$ .

Factor $3$ out of $3 x$ .

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

Factor $3$ out of $- 3$ .

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

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

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

Factor $3$ out of $3$ .

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

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

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $x - 1 + 1$ by $1$ .

$f (3 x - 3) = x - 1 + 1$

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

Add $- 1$ and $1$ .

$f (3 x - 3) = x + 0$

Add $x$ and $0$ .

$f (3 x - 3) = x$

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

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

Find the Inverse y=(x+3)/3

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