Trigonometry Examples

$y = \frac{1}{3^{x}}$

Step 1

Interchange the variables.

$x = \frac{1}{3^{y}}$

Step 2

Solve for $y$ .

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Step 2.1

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

$\frac{1}{3^{y}} = x$

Step 2.2

Multiply both sides by $3^{y}$ .

$\frac{1}{3^{y}} \cdot 3^{y} = x \cdot 3^{y}$

Step 2.3

Simplify the left side.

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Step 2.3.1

Cancel the common factor of $3^{y}$ .

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Step 2.3.1.1

Cancel the common factor.

$\frac{1}{3^{y}} \cdot 3^{y} = x \cdot 3^{y}$

Step 2.3.1.2

Rewrite the expression.

$1 = x \cdot 3^{y}$

Step 2.4

Solve for $y$ .

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Step 2.4.1

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

$x \cdot 3^{y} = 1$

Step 2.4.2

Divide each term in $x \cdot 3^{y} = 1$ by $x$ and simplify.

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Step 2.4.2.1

Divide each term in $x \cdot 3^{y} = 1$ by $x$ .

$\frac{x \cdot 3^{y}}{x} = \frac{1}{x}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $x$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{x \cdot 3^{y}}{x} = \frac{1}{x}$

Step 2.4.2.2.1.2

Divide $3^{y}$ by $1$ .

$3^{y} = \frac{1}{x}$

Step 2.4.3

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

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

Step 2.4.4

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

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

Step 2.4.5

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

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Step 2.4.5.1

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

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

Step 2.4.5.2

Simplify the left side.

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Step 2.4.5.2.1

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

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Step 2.4.5.2.1.1

Cancel the common factor.

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

Step 2.4.5.2.1.2

Divide $y$ by $1$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

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

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2.3.2

Multiply $3^{x}$ by $1$ .

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

Step 4.2.4

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

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

Step 4.2.5

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

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Step 4.2.5.1

Cancel the common factor.

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

Step 4.2.5.2

Divide $x$ by $1$ .

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

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

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

Step 4.3.3

Simplify the denominator.

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Step 4.3.3.1

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

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

Step 4.3.3.2

Exponentiation and log are inverse functions.

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

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.5

Multiply $x$ by $1$ .

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

Step 4.4

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

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