Algebra Examples

$f (x) = 12 \cdot 11^{3 x} + 13$

Step 1

Write $f (x) = 12 \cdot 11^{3 x} + 13$ as an equation.

$y = 12 \cdot 11^{3 x} + 13$

Step 2

Interchange the variables.

$x = 12 \cdot 11^{3 y} + 13$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $12 \cdot 11^{3 y} + 13 = x$ .

$12 \cdot 11^{3 y} + 13 = x$

Step 3.2

Subtract $13$ from both sides of the equation.

$12 \cdot 11^{3 y} = x - 13$

Step 3.3

Divide each term in $12 \cdot 11^{3 y} = x - 13$ by $12$ and simplify.

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Step 3.3.1

Divide each term in $12 \cdot 11^{3 y} = x - 13$ by $12$ .

$\frac{12 \cdot 11^{3 y}}{12} = \frac{x}{12} + \frac{- 13}{12}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $12$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{12 \cdot 11^{3 y}}{12} = \frac{x}{12} + \frac{- 13}{12}$

Step 3.3.2.1.2

Divide $11^{3 y}$ by $1$ .

$11^{3 y} = \frac{x}{12} + \frac{- 13}{12}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$11^{3 y} = \frac{x}{12} - \frac{13}{12}$

Step 3.4

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

$\ln (11^{3 y}) = \ln (\frac{x}{12} - \frac{13}{12})$

Step 3.5

Expand $\ln (11^{3 y})$ by moving $3 y$ outside the logarithm.

$3 y \ln (11) = \ln (\frac{x}{12} - \frac{13}{12})$

Step 3.6

Divide each term in $3 y \ln (11) = \ln (\frac{x}{12} - \frac{13}{12})$ by $3 \ln (11)$ and simplify.

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Step 3.6.1

Divide each term in $3 y \ln (11) = \ln (\frac{x}{12} - \frac{13}{12})$ by $3 \ln (11)$ .

$\frac{3 y \ln (11)}{3 \ln (11)} = \frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $3$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{3 y \ln (11)}{3 \ln (11)} = \frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}$

Step 3.6.2.1.2

Rewrite the expression.

$\frac{y \ln (11)}{\ln (11)} = \frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}$

Step 3.6.2.2

Cancel the common factor of $\ln (11)$ .

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Step 3.6.2.2.1

Cancel the common factor.

$\frac{y \ln (11)}{\ln (11)} = \frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}$

Step 3.6.2.2.2

Divide $y$ by $1$ .

$y = \frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}$

Step 4

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

$f^{- 1} (x) = \frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}$

Step 5

Verify if $f^{- 1} (x) = \frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}$ is the inverse of $f (x) = 12 \cdot 11^{3 x} + 13$ .

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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 $f^{- 1} (12 \cdot 11^{3 x} + 13)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (12 \cdot 11^{3 x} + 13) = \frac{\ln (\frac{12 \cdot 11^{3 x} + 13}{12} - \frac{13}{12})}{3 \ln (11)}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Combine the numerators over the common denominator.

$f^{- 1} (12 \cdot 11^{3 x} + 13) = \frac{\ln (\frac{12 \cdot 11^{3 x} + 13 - 13}{12})}{3 \ln (11)}$

Step 5.2.3.2

Combine the opposite terms in $12 \cdot 11^{3 x} + 13 - 13$ .

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Step 5.2.3.2.1

Subtract $13$ from $13$ .

$f^{- 1} (12 \cdot 11^{3 x} + 13) = \frac{\ln (\frac{12 \cdot 11^{3 x} + 0}{12})}{3 \ln (11)}$

Step 5.2.3.2.2

Add $12 \cdot 11^{3 x}$ and $0$ .

$f^{- 1} (12 \cdot 11^{3 x} + 13) = \frac{\ln (\frac{12 \cdot 11^{3 x}}{12})}{3 \ln (11)}$

Step 5.2.3.3

Cancel the common factor of $12$ .

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Step 5.2.3.3.1

Cancel the common factor.

$f^{- 1} (12 \cdot 11^{3 x} + 13) = \frac{\ln (\frac{12 \cdot 11^{3 x}}{12})}{3 \ln (11)}$

Step 5.2.3.3.2

Divide $11^{3 x}$ by $1$ .

$f^{- 1} (12 \cdot 11^{3 x} + 13) = \frac{\ln (11^{3 x})}{3 \ln (11)}$

Step 5.2.4

Simplify $3 \ln (11)$ by moving $3$ inside the logarithm.

$f^{- 1} (12 \cdot 11^{3 x} + 13) = \frac{\ln (11^{3 x})}{\ln (11^{3})}$

Step 5.2.5

Raise $11$ to the power of $3$ .

$f^{- 1} (12 \cdot 11^{3 x} + 13) = \frac{\ln (11^{3 x})}{\ln (1331)}$

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 $f (\frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}) = 12 \cdot 11^{3 (\frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)})} + 13$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Combine the numerators over the common denominator.

$f (\frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}) = 12 \cdot 11^{3 (\frac{\ln (\frac{x - 13}{12})}{3 \ln (11)})} + 13$

Step 5.3.3.2

Simplify $3 \ln (11)$ by moving $3$ inside the logarithm.

$f (\frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}) = 12 \cdot 11^{3 (\frac{\ln (\frac{x - 13}{12})}{\ln (11^{3})})} + 13$

Step 5.3.3.3

Raise $11$ to the power of $3$ .

$f (\frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}) = 12 \cdot 11^{3 (\frac{\ln (\frac{x - 13}{12})}{\ln (1331)})} + 13$

Step 5.3.3.4

Multiply $3 \frac{\ln (\frac{x - 13}{12})}{\ln (1331)}$ .

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Step 5.3.3.4.1

Combine $3$ and $\frac{\ln (\frac{x - 13}{12})}{\ln (1331)}$ .

$f (\frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}) = 12 \cdot 11^{\frac{3 \ln (\frac{x - 13}{12})}{\ln (1331)}} + 13$

Step 5.3.3.4.2

Simplify $3 \ln (\frac{x - 13}{12})$ by moving $3$ inside the logarithm.

$f (\frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}) = 12 \cdot 11^{\frac{\ln ({(\frac{x - 13}{12})}^{3})}{\ln (1331)}} + 13$

Step 5.3.3.5

Simplify the numerator.

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Step 5.3.3.5.1

Apply the product rule to $\frac{x - 13}{12}$ .

$f (\frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}) = 12 \cdot 11^{\frac{\ln (\frac{{(x - 13)}^{3}}{12^{3}})}{\ln (1331)}} + 13$

Step 5.3.3.5.2

Raise $12$ to the power of $3$ .

$f (\frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}) = 12 \cdot 11^{\frac{\ln (\frac{{(x - 13)}^{3}}{1728})}{\ln (1331)}} + 13$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}$ is the inverse of $f (x) = 12 \cdot 11^{3 x} + 13$ .

$f^{- 1} (x) = \frac{\ln (\frac{x}{12} - \frac{13}{12})}{3 \ln (11)}$