Trigonometry Examples

$f (x) = 6^{x} - 3$

Step 1

Write $f (x) = 6^{x} - 3$ as an equation.

$y = 6^{x} - 3$

Step 2

Interchange the variables.

$x = 6^{y} - 3$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $6^{y} - 3 = x$ .

$6^{y} - 3 = x$

Step 3.2

Add $3$ to both sides of the equation.

$6^{y} = x + 3$

Step 3.3

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

$\ln (6^{y}) = \ln (x + 3)$

Step 3.4

Expand $\ln (6^{y})$ by moving $y$ outside the logarithm.

$y \ln (6) = \ln (x + 3)$

Step 3.5

Divide each term in $y \ln (6) = \ln (x + 3)$ by $\ln (6)$ and simplify.

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Step 3.5.1

Divide each term in $y \ln (6) = \ln (x + 3)$ by $\ln (6)$ .

$\frac{y \ln (6)}{\ln (6)} = \frac{\ln (x + 3)}{\ln (6)}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $\ln (6)$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{y \ln (6)}{\ln (6)} = \frac{\ln (x + 3)}{\ln (6)}$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (x + 3)}{\ln (6)}$

Step 4

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

$f^{- 1} (x) = \frac{\ln (x + 3)}{\ln (6)}$

Step 5

Verify if $f^{- 1} (x) = \frac{\ln (x + 3)}{\ln (6)}$ is the inverse of $f (x) = 6^{x} - 3$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (6^{x} - 3)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (6^{x} - 3) = \frac{\ln ((6^{x} - 3) + 3)}{\ln (6)}$

Step 5.2.3

Combine the opposite terms in $6^{x} - 3 + 3$ .

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Step 5.2.3.1

Add $- 3$ and $3$ .

$f^{- 1} (6^{x} - 3) = \frac{\ln (6^{x} + 0)}{\ln (6)}$

Step 5.2.3.2

Add $6^{x}$ and $0$ .

$f^{- 1} (6^{x} - 3) = \frac{\ln (6^{x})}{\ln (6)}$

Step 5.2.4

Expand $\ln (6^{x})$ by moving $x$ outside the logarithm.

$f^{- 1} (6^{x} - 3) = \frac{x \ln (6)}{\ln (6)}$

Step 5.2.5

Cancel the common factor of $\ln (6)$ .

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Step 5.2.5.1

Cancel the common factor.

$f^{- 1} (6^{x} - 3) = \frac{x \ln (6)}{\ln (6)}$

Step 5.2.5.2

Divide $x$ by $1$ .

$f^{- 1} (6^{x} - 3) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{\ln (x + 3)}{\ln (6)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\ln (x + 3)}{\ln (6)}) = 6^{\frac{\ln (x + 3)}{\ln (6)}} - 3$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

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

$f (\frac{\ln (x + 3)}{\ln (6)}) = 6^{\log_{6} (x + 3)} - 3$

Step 5.3.3.2

Exponentiation and log are inverse functions.

$f (\frac{\ln (x + 3)}{\ln (6)}) = x + 3 - 3$

Step 5.3.4

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

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Step 5.3.4.1

Subtract $3$ from $3$ .

$f (\frac{\ln (x + 3)}{\ln (6)}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{\ln (x + 3)}{\ln (6)}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\ln (x + 3)}{\ln (6)}$ is the inverse of $f (x) = 6^{x} - 3$ .

$f^{- 1} (x) = \frac{\ln (x + 3)}{\ln (6)}$