Precalculus Examples

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

Step 1

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

$y = 3 x$

Step 2

Interchange the variables.

$x = 3 y$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $3 y = x$ .

$3 y = x$

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide $y$ by $1$ .

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

Step 4

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

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

Step 5

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

Tap for more steps...

Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

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

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

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

Step 5.2.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.3.1

Cancel the common factor.

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

Step 5.2.3.2

Divide $x$ by $1$ .

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

Step 5.3

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

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

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

Step 5.3.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.3.3.1

Cancel the common factor.

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

Step 5.3.3.2

Rewrite the expression.

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

Step 5.4

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

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