Trigonometry Examples

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

Step 1

Write $f (x) = 3^{x} - 1$ as an equation.

$y = 3^{x} - 1$

Step 2

Interchange the variables.

$x = 3^{y} - 1$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $3^{y} - 1 = x$ .

$3^{y} - 1 = x$

Step 3.2

Add $1$ to both sides of the equation.

$3^{y} = x + 1$

Step 3.3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (3^{y}) = \ln (x + 1)$

Step 3.4

Expand $\ln (3^{y})$ by moving $y$ outside the logarithm.

$y \ln (3) = \ln (x + 1)$

Step 3.5

Divide each term in $y \ln (3) = \ln (x + 1)$ by $\ln (3)$ and simplify.

Tap for more steps...

Step 3.5.1

Divide each term in $y \ln (3) = \ln (x + 1)$ by $\ln (3)$ .

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

Step 3.5.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.1

Cancel the common factor of $\ln (3)$ .

Tap for more steps...

Step 3.5.2.1.1

Cancel the common factor.

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

Step 3.5.2.1.2

Divide $y$ by $1$ .

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

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

Tap for more steps...

Step 5.2.3.1

Add $- 1$ and $1$ .

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

Step 5.2.3.2

Add $3^{x}$ and $0$ .

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

Step 5.2.4

Expand $\ln (3^{x})$ by moving $x$ outside the logarithm.

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

Step 5.2.5

Cancel the common factor of $\ln (3)$ .

Tap for more steps...

Step 5.2.5.1

Cancel the common factor.

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

Step 5.2.5.2

Divide $x$ by $1$ .

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

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

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

Step 5.3.3

Simplify each term.

Tap for more steps...

Step 5.3.3.1

Use the change of base rule $\frac{\log_{b} (x)}{\log_{b} (a)} = \log_{a} (x)$ .

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

Step 5.3.3.2

Exponentiation and log are inverse functions.

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

Step 5.3.4

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

Tap for more steps...

Step 5.3.4.1

Subtract $1$ from $1$ .

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

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

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

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