Algebra Examples

$f (x) = 2 \cdot 3^{2 x} + 4$

Step 1

Write $f (x) = 2 \cdot 3^{2 x} + 4$ as an equation.

$y = 2 \cdot 3^{2 x} + 4$

Step 2

Interchange the variables.

$x = 2 \cdot 3^{2 y} + 4$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $2 \cdot 3^{2 y} + 4 = x$ .

$2 \cdot 3^{2 y} + 4 = x$

Step 3.2

Subtract $4$ from both sides of the equation.

$2 \cdot 3^{2 y} = x - 4$

Step 3.3

Divide each term in $2 \cdot 3^{2 y} = x - 4$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 \cdot 3^{2 y} = x - 4$ by $2$ .

$\frac{2 \cdot 3^{2 y}}{2} = \frac{x}{2} + \frac{- 4}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 \cdot 3^{2 y}}{2} = \frac{x}{2} + \frac{- 4}{2}$

Step 3.3.2.1.2

Divide $3^{2 y}$ by $1$ .

$3^{2 y} = \frac{x}{2} + \frac{- 4}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 4$ by $2$ .

$3^{2 y} = \frac{x}{2} - 2$

Step 3.4

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

$\ln (3^{2 y}) = \ln (\frac{x}{2} - 2)$

Step 3.5

Expand $\ln (3^{2 y})$ by moving $2 y$ outside the logarithm.

$2 y \ln (3) = \ln (\frac{x}{2} - 2)$

Step 3.6

Divide each term in $2 y \ln (3) = \ln (\frac{x}{2} - 2)$ by $2 \ln (3)$ and simplify.

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Step 3.6.1

Divide each term in $2 y \ln (3) = \ln (\frac{x}{2} - 2)$ by $2 \ln (3)$ .

$\frac{2 y \ln (3)}{2 \ln (3)} = \frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $2$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{2 y \ln (3)}{2 \ln (3)} = \frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}$

Step 3.6.2.1.2

Rewrite the expression.

$\frac{y \ln (3)}{\ln (3)} = \frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}$

Step 3.6.2.2

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

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Step 3.6.2.2.1

Cancel the common factor.

$\frac{y \ln (3)}{\ln (3)} = \frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}$

Step 3.6.2.2.2

Divide $y$ by $1$ .

$y = \frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}$

Step 4

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

$f^{- 1} (x) = \frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}$

Step 5

Verify if $f^{- 1} (x) = \frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}$ is the inverse of $f (x) = 2 \cdot 3^{2 x} + 4$ .

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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 $f^{- 1} (2 \cdot 3^{2 x} + 4)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (2 \cdot 3^{2 x} + 4) = \frac{\ln (\frac{2 \cdot 3^{2 x} + 4}{2} - 2)}{2 \ln (3)}$

Step 5.2.3

Cancel the common factor of $2 \cdot 3^{2 x} + 4$ and $2$ .

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Step 5.2.3.1

Factor $2$ out of $2 \cdot 3^{2 x}$ .

$f^{- 1} (2 \cdot 3^{2 x} + 4) = \frac{\ln (\frac{2 (3^{2 x}) + 4}{2} - 2)}{2 \ln (3)}$

Step 5.2.3.2

Factor $2$ out of $4$ .

$f^{- 1} (2 \cdot 3^{2 x} + 4) = \frac{\ln (\frac{2 (3^{2 x}) + 2 \cdot 2}{2} - 2)}{2 \ln (3)}$

Step 5.2.3.3

Factor $2$ out of $2 (3^{2 x}) + 2 (2)$ .

$f^{- 1} (2 \cdot 3^{2 x} + 4) = \frac{\ln (\frac{2 (3^{2 x} + 2)}{2} - 2)}{2 \ln (3)}$

Step 5.2.3.4

Cancel the common factors.

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Step 5.2.3.4.1

Factor $2$ out of $2$ .

$f^{- 1} (2 \cdot 3^{2 x} + 4) = \frac{\ln (\frac{2 (3^{2 x} + 2)}{2 (1)} - 2)}{2 \ln (3)}$

Step 5.2.3.4.2

Cancel the common factor.

$f^{- 1} (2 \cdot 3^{2 x} + 4) = \frac{\ln (\frac{2 (3^{2 x} + 2)}{2 \cdot 1} - 2)}{2 \ln (3)}$

Step 5.2.3.4.3

Rewrite the expression.

$f^{- 1} (2 \cdot 3^{2 x} + 4) = \frac{\ln (\frac{3^{2 x} + 2}{1} - 2)}{2 \ln (3)}$

Step 5.2.3.4.4

Divide $3^{2 x} + 2$ by $1$ .

$f^{- 1} (2 \cdot 3^{2 x} + 4) = \frac{\ln (3^{2 x} + 2 - 2)}{2 \ln (3)}$

Step 5.2.4

Combine the opposite terms in $3^{2 x} + 2 - 2$ .

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Step 5.2.4.1

Subtract $2$ from $2$ .

$f^{- 1} (2 \cdot 3^{2 x} + 4) = \frac{\ln (3^{2 x} + 0)}{2 \ln (3)}$

Step 5.2.4.2

Add $3^{2 x}$ and $0$ .

$f^{- 1} (2 \cdot 3^{2 x} + 4) = \frac{\ln (3^{2 x})}{2 \ln (3)}$

Step 5.2.5

Simplify $2 \ln (3)$ by moving $2$ inside the logarithm.

$f^{- 1} (2 \cdot 3^{2 x} + 4) = \frac{\ln (3^{2 x})}{\ln (3^{2})}$

Step 5.2.6

Raise $3$ to the power of $2$ .

$f^{- 1} (2 \cdot 3^{2 x} + 4) = \frac{\ln (3^{2 x})}{\ln (9)}$

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 $f (\frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}) = 2 \cdot 3^{2 (\frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)})} + 4$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Simplify the numerator.

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Step 5.3.3.1.1

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}) = 2 \cdot 3^{2 (\frac{\ln (\frac{x}{2} - 2 \cdot \frac{2}{2})}{2 \ln (3)})} + 4$

Step 5.3.3.1.2

Combine $- 2$ and $\frac{2}{2}$ .

$f (\frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}) = 2 \cdot 3^{2 (\frac{\ln (\frac{x}{2} + \frac{- 2 \cdot 2}{2})}{2 \ln (3)})} + 4$

Step 5.3.3.1.3

Combine the numerators over the common denominator.

$f (\frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}) = 2 \cdot 3^{2 (\frac{\ln (\frac{x - 2 \cdot 2}{2})}{2 \ln (3)})} + 4$

Step 5.3.3.1.4

Multiply $- 2$ by $2$ .

$f (\frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}) = 2 \cdot 3^{2 (\frac{\ln (\frac{x - 4}{2})}{2 \ln (3)})} + 4$

Step 5.3.3.2

Simplify $2 \ln (3)$ by moving $2$ inside the logarithm.

$f (\frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}) = 2 \cdot 3^{2 (\frac{\ln (\frac{x - 4}{2})}{\ln (3^{2})})} + 4$

Step 5.3.3.3

Raise $3$ to the power of $2$ .

$f (\frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}) = 2 \cdot 3^{2 (\frac{\ln (\frac{x - 4}{2})}{\ln (9)})} + 4$

Step 5.3.3.4

Multiply $2 \frac{\ln (\frac{x - 4}{2})}{\ln (9)}$ .

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Step 5.3.3.4.1

Combine $2$ and $\frac{\ln (\frac{x - 4}{2})}{\ln (9)}$ .

$f (\frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}) = 2 \cdot 3^{\frac{2 \ln (\frac{x - 4}{2})}{\ln (9)}} + 4$

Step 5.3.3.4.2

Simplify $2 \ln (\frac{x - 4}{2})$ by moving $2$ inside the logarithm.

$f (\frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}) = 2 \cdot 3^{\frac{\ln ({(\frac{x - 4}{2})}^{2})}{\ln (9)}} + 4$

Step 5.3.3.5

Simplify the numerator.

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Step 5.3.3.5.1

Apply the product rule to $\frac{x - 4}{2}$ .

$f (\frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}) = 2 \cdot 3^{\frac{\ln (\frac{{(x - 4)}^{2}}{2^{2}})}{\ln (9)}} + 4$

Step 5.3.3.5.2

Raise $2$ to the power of $2$ .

$f (\frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}) = 2 \cdot 3^{\frac{\ln (\frac{{(x - 4)}^{2}}{4})}{\ln (9)}} + 4$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}$ is the inverse of $f (x) = 2 \cdot 3^{2 x} + 4$ .

$f^{- 1} (x) = \frac{\ln (\frac{x}{2} - 2)}{2 \ln (3)}$