Trigonometry Examples

$f (x) = \frac{3 e^{x} - 8}{20 e^{x} + 15}$

Step 1

Write $f (x) = \frac{3 e^{x} - 8}{20 e^{x} + 15}$ as an equation.

$y = \frac{3 e^{x} - 8}{20 e^{x} + 15}$

Step 2

Interchange the variables.

$x = \frac{3 e^{y} - 8}{20 e^{y} + 15}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{3 e^{y} - 8}{20 e^{y} + 15} = x$ .

$\frac{3 e^{y} - 8}{20 e^{y} + 15} = x$

Step 3.2

Multiply both sides by $20 e^{y} + 15$ .

$\frac{3 e^{y} - 8}{20 e^{y} + 15} (20 e^{y} + 15) = x (20 e^{y} + 15)$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Cancel the common factor of $20 e^{y} + 15$ .

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Step 3.3.1.1.1

Cancel the common factor.

$\frac{3 e^{y} - 8}{20 e^{y} + 15} (20 e^{y} + 15) = x (20 e^{y} + 15)$

Step 3.3.1.1.2

Rewrite the expression.

$3 e^{y} - 8 = x (20 e^{y} + 15)$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $x (20 e^{y} + 15)$ .

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Step 3.3.2.1.1

Apply the distributive property.

$3 e^{y} - 8 = x (20 e^{y}) + x \cdot 15$

Step 3.3.2.1.2

Reorder.

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Step 3.3.2.1.2.1

Rewrite using the commutative property of multiplication.

$3 e^{y} - 8 = 20 x e^{y} + x \cdot 15$

Step 3.3.2.1.2.2

Move $15$ to the left of $x$ .

$3 e^{y} - 8 = 20 x e^{y} + 15 x$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $20 x e^{y}$ from both sides of the equation.

$3 e^{y} - 8 - 20 x e^{y} = 15 x$

Step 3.4.2

Add $8$ to both sides of the equation.

$3 e^{y} - 20 x e^{y} = 15 x + 8$

Step 3.4.3

Factor $e^{y}$ out of $3 e^{y} - 20 x e^{y}$ .

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Step 3.4.3.1

Factor $e^{y}$ out of $3 e^{y}$ .

$e^{y} \cdot 3 - 20 x e^{y} = 15 x + 8$

Step 3.4.3.2

Factor $e^{y}$ out of $- 20 x e^{y}$ .

$e^{y} \cdot 3 + e^{y} (- 20 x) = 15 x + 8$

Step 3.4.3.3

Factor $e^{y}$ out of $e^{y} \cdot 3 + e^{y} (- 20 x)$ .

$e^{y} (3 - 20 x) = 15 x + 8$

Step 3.4.4

Divide each term in $e^{y} (3 - 20 x) = 15 x + 8$ by $3 - 20 x$ and simplify.

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Step 3.4.4.1

Divide each term in $e^{y} (3 - 20 x) = 15 x + 8$ by $3 - 20 x$ .

$\frac{e^{y} (3 - 20 x)}{3 - 20 x} = \frac{15 x}{3 - 20 x} + \frac{8}{3 - 20 x}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $3 - 20 x$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{e^{y} (3 - 20 x)}{3 - 20 x} = \frac{15 x}{3 - 20 x} + \frac{8}{3 - 20 x}$

Step 3.4.4.2.1.2

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

$e^{y} = \frac{15 x}{3 - 20 x} + \frac{8}{3 - 20 x}$

Step 3.4.4.3

Simplify the right side.

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Step 3.4.4.3.1

Combine the numerators over the common denominator.

$e^{y} = \frac{15 x + 8}{3 - 20 x}$

Step 3.4.5

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

$\ln (e^{y}) = \ln (\frac{15 x + 8}{3 - 20 x})$

Step 3.4.6

Expand the left side.

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Step 3.4.6.1

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

$y \ln (e) = \ln (\frac{15 x + 8}{3 - 20 x})$

Step 3.4.6.2

The natural logarithm of $e$ is $1$ .

$y \cdot 1 = \ln (\frac{15 x + 8}{3 - 20 x})$

Step 3.4.6.3

Multiply $y$ by $1$ .

$y = \ln (\frac{15 x + 8}{3 - 20 x})$

Step 4

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

$f^{- 1} (x) = \ln (\frac{15 x + 8}{3 - 20 x})$

Step 5

Verify if $f^{- 1} (x) = \ln (\frac{15 x + 8}{3 - 20 x})$ is the inverse of $f (x) = \frac{3 e^{x} - 8}{20 e^{x} + 15}$ .

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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 $f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{15 (\frac{3 e^{x} - 8}{20 e^{x} + 15}) + 8}{3 - 20 \frac{3 e^{x} - 8}{20 e^{x} + 15}})$

Step 5.2.3

Simplify with factoring out.

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Step 5.2.3.1

Factor $5$ out of $20 e^{x} + 15$ .

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Step 5.2.3.1.1

Factor $5$ out of $20 e^{x}$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{15 (\frac{3 e^{x} - 8}{5 (4 e^{x}) + 15}) + 8}{3 - 20 \frac{3 e^{x} - 8}{20 e^{x} + 15}})$

Step 5.2.3.1.2

Factor $5$ out of $15$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{15 (\frac{3 e^{x} - 8}{5 (4 e^{x}) + 5 (3)}) + 8}{3 - 20 \frac{3 e^{x} - 8}{20 e^{x} + 15}})$

Step 5.2.3.1.3

Factor $5$ out of $5 (4 e^{x}) + 5 (3)$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{15 (\frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}) + 8}{3 - 20 \frac{3 e^{x} - 8}{20 e^{x} + 15}})$

Step 5.2.3.2

Factor $5$ out of $20 e^{x} + 15$ .

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Step 5.2.3.2.1

Factor $5$ out of $20 e^{x}$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{15 (\frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}) + 8}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x}) + 15}})$

Step 5.2.3.2.2

Factor $5$ out of $15$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{15 (\frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}) + 8}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x}) + 5 (3)}})$

Step 5.2.3.2.3

Factor $5$ out of $5 (4 e^{x}) + 5 (3)$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{15 (\frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}) + 8}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

Cancel the common factor of $5$ .

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Step 5.2.4.1.1

Factor $5$ out of $15$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{5 (3) (\frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}) + 8}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.4.1.2

Cancel the common factor.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{5 \cdot (3 (\frac{3 e^{x} - 8}{5 (4 e^{x} + 3)})) + 8}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.4.1.3

Rewrite the expression.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{3 (\frac{3 e^{x} - 8}{4 e^{x} + 3}) + 8}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.4.2

Combine $3$ and $\frac{3 e^{x} - 8}{4 e^{x} + 3}$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{3 (3 e^{x} - 8)}{4 e^{x} + 3} + 8}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.4.3

To write $8$ as a fraction with a common denominator, multiply by $\frac{4 e^{x} + 3}{4 e^{x} + 3}$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{3 (3 e^{x} - 8)}{4 e^{x} + 3} + \frac{8 (4 e^{x} + 3)}{4 e^{x} + 3}}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.4.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{3 (3 e^{x} - 8) + 8 (4 e^{x} + 3)}{4 e^{x} + 3}}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.4.5

Reorder terms.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{8 (4 e^{x} + 3) + 3 (3 e^{x} - 8)}{4 e^{x} + 3}}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.4.6

Rewrite $\frac{8 (4 e^{x} + 3) + 3 (3 e^{x} - 8)}{4 e^{x} + 3}$ in a factored form.

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Step 5.2.4.6.1

Apply the distributive property.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{8 (4 e^{x}) + 8 \cdot 3 + 3 (3 e^{x} - 8)}{4 e^{x} + 3}}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.4.6.2

Multiply $4$ by $8$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{32 e^{x} + 8 \cdot 3 + 3 (3 e^{x} - 8)}{4 e^{x} + 3}}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.4.6.3

Multiply $8$ by $3$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{32 e^{x} + 24 + 3 (3 e^{x} - 8)}{4 e^{x} + 3}}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.4.6.4

Apply the distributive property.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{32 e^{x} + 24 + 3 (3 e^{x}) + 3 \cdot - 8}{4 e^{x} + 3}}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.4.6.5

Multiply $3$ by $3$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{32 e^{x} + 24 + 9 e^{x} + 3 \cdot - 8}{4 e^{x} + 3}}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.4.6.6

Multiply $3$ by $- 8$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{32 e^{x} + 24 + 9 e^{x} - 24}{4 e^{x} + 3}}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.4.6.7

Add $32 e^{x}$ and $9 e^{x}$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x} + 24 - 24}{4 e^{x} + 3}}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.4.6.8

Subtract $24$ from $24$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x} + 0}{4 e^{x} + 3}}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.4.6.9

Add $41 e^{x}$ and $0$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{3 - 20 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)}})$

Step 5.2.5

Simplify the denominator.

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Step 5.2.5.1

Cancel the common factor of $5$ .

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Step 5.2.5.1.1

Factor $5$ out of $- 20$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{3 + 5 (- 4) (\frac{3 e^{x} - 8}{5 (4 e^{x} + 3)})})$

Step 5.2.5.1.2

Cancel the common factor.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{3 + 5 \cdot (- 4 \frac{3 e^{x} - 8}{5 (4 e^{x} + 3)})})$

Step 5.2.5.1.3

Rewrite the expression.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{3 - 4 \frac{3 e^{x} - 8}{4 e^{x} + 3}})$

Step 5.2.5.2

Combine $- 4$ and $\frac{3 e^{x} - 8}{4 e^{x} + 3}$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{3 + \frac{- 4 (3 e^{x} - 8)}{4 e^{x} + 3}})$

Step 5.2.5.3

Move the negative in front of the fraction.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{3 - \frac{4 (3 e^{x} - 8)}{4 e^{x} + 3}})$

Step 5.2.5.4

To write $3$ as a fraction with a common denominator, multiply by $\frac{4 e^{x} + 3}{4 e^{x} + 3}$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{\frac{3 (4 e^{x} + 3)}{4 e^{x} + 3} - \frac{4 (3 e^{x} - 8)}{4 e^{x} + 3}})$

Step 5.2.5.5

Combine the numerators over the common denominator.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{\frac{3 (4 e^{x} + 3) - 4 (3 e^{x} - 8)}{4 e^{x} + 3}})$

Step 5.2.5.6

Rewrite $\frac{3 (4 e^{x} + 3) - 4 (3 e^{x} - 8)}{4 e^{x} + 3}$ in a factored form.

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Step 5.2.5.6.1

Apply the distributive property.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{\frac{3 (4 e^{x}) + 3 \cdot 3 - 4 (3 e^{x} - 8)}{4 e^{x} + 3}})$

Step 5.2.5.6.2

Multiply $4$ by $3$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{\frac{12 e^{x} + 3 \cdot 3 - 4 (3 e^{x} - 8)}{4 e^{x} + 3}})$

Step 5.2.5.6.3

Multiply $3$ by $3$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{\frac{12 e^{x} + 9 - 4 (3 e^{x} - 8)}{4 e^{x} + 3}})$

Step 5.2.5.6.4

Apply the distributive property.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{\frac{12 e^{x} + 9 - 4 (3 e^{x}) - 4 \cdot - 8}{4 e^{x} + 3}})$

Step 5.2.5.6.5

Multiply $3$ by $- 4$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{\frac{12 e^{x} + 9 - 12 e^{x} - 4 \cdot - 8}{4 e^{x} + 3}})$

Step 5.2.5.6.6

Multiply $- 4$ by $- 8$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{\frac{12 e^{x} + 9 - 12 e^{x} + 32}{4 e^{x} + 3}})$

Step 5.2.5.6.7

Subtract $12 e^{x}$ from $12 e^{x}$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{\frac{0 + 9 + 32}{4 e^{x} + 3}})$

Step 5.2.5.6.8

Add $0$ and $9$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{\frac{9 + 32}{4 e^{x} + 3}})$

Step 5.2.5.6.9

Add $9$ and $32$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{\frac{41 e^{x}}{4 e^{x} + 3}}{\frac{41}{4 e^{x} + 3}})$

Step 5.2.6

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{41 e^{x}}{4 e^{x} + 3} \cdot \frac{4 e^{x} + 3}{41})$

Step 5.2.7

Cancel the common factor of $41$ .

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Step 5.2.7.1

Factor $41$ out of $41 e^{x}$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{41 (e^{x})}{4 e^{x} + 3} \cdot \frac{4 e^{x} + 3}{41})$

Step 5.2.7.2

Cancel the common factor.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{41 e^{x}}{4 e^{x} + 3} \cdot \frac{4 e^{x} + 3}{41})$

Step 5.2.7.3

Rewrite the expression.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{e^{x}}{4 e^{x} + 3} \cdot (4 e^{x} + 3))$

Step 5.2.8

Cancel the common factor of $4 e^{x} + 3$ .

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Step 5.2.8.1

Cancel the common factor.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (\frac{e^{x}}{4 e^{x} + 3} \cdot (4 e^{x} + 3))$

Step 5.2.8.2

Rewrite the expression.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = \ln (e^{x})$

Step 5.2.9

Use logarithm rules to move $x$ out of the exponent.

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = x \ln (e)$

Step 5.2.10

The natural logarithm of $e$ is $1$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = x \cdot 1$

Step 5.2.11

Multiply $x$ by $1$ .

$f^{- 1} (\frac{3 e^{x} - 8}{20 e^{x} + 15}) = x$

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 $f (\ln (\frac{15 x + 8}{3 - 20 x}))$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{3 e^{\ln (\frac{15 x + 8}{3 - 20 x})} - 8}{20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Exponentiation and log are inverse functions.

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{3 (\frac{15 x + 8}{3 - 20 x}) - 8}{20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15}$

Step 5.3.3.2

Combine $3$ and $\frac{15 x + 8}{3 - 20 x}$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{3 (15 x + 8)}{3 - 20 x} - 8}{20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15}$

Step 5.3.3.3

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{3 - 20 x}{3 - 20 x}$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{3 (15 x + 8)}{3 - 20 x} - 8 \cdot \frac{3 - 20 x}{3 - 20 x}}{20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15}$

Step 5.3.3.4

Combine $- 8$ and $\frac{3 - 20 x}{3 - 20 x}$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{3 (15 x + 8)}{3 - 20 x} + \frac{- 8 (3 - 20 x)}{3 - 20 x}}{20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15}$

Step 5.3.3.5

Combine the numerators over the common denominator.

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{3 (15 x + 8) - 8 (3 - 20 x)}{3 - 20 x}}{20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15}$

Step 5.3.3.6

Reorder terms.

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{3 (15 x + 8) - 8 (3 - 20 x)}{- 20 x + 3}}{20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15}$

Step 5.3.3.7

Rewrite $\frac{3 (15 x + 8) - 8 (3 - 20 x)}{- 20 x + 3}$ in a factored form.

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Step 5.3.3.7.1

Apply the distributive property.

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{3 (15 x) + 3 \cdot 8 - 8 (3 - 20 x)}{- 20 x + 3}}{20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15}$

Step 5.3.3.7.2

Multiply $15$ by $3$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{45 x + 3 \cdot 8 - 8 (3 - 20 x)}{- 20 x + 3}}{20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15}$

Step 5.3.3.7.3

Multiply $3$ by $8$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{45 x + 24 - 8 (3 - 20 x)}{- 20 x + 3}}{20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15}$

Step 5.3.3.7.4

Apply the distributive property.

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{45 x + 24 - 8 \cdot 3 - 8 (- 20 x)}{- 20 x + 3}}{20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15}$

Step 5.3.3.7.5

Multiply $- 8$ by $3$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{45 x + 24 - 24 - 8 (- 20 x)}{- 20 x + 3}}{20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15}$

Step 5.3.3.7.6

Multiply $- 20$ by $- 8$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{45 x + 24 - 24 + 160 x}{- 20 x + 3}}{20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15}$

Step 5.3.3.7.7

Add $45 x$ and $160 x$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x + 24 - 24}{- 20 x + 3}}{20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15}$

Step 5.3.3.7.8

Subtract $24$ from $24$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x + 0}{- 20 x + 3}}{20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15}$

Step 5.3.3.7.9

Add $205 x$ and $0$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15}$

Step 5.3.4

Simplify the denominator.

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Step 5.3.4.1

Factor $5$ out of $20 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 15$ .

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Step 5.3.4.1.1

Factor $5$ out of $20 e^{\ln (\frac{15 x + 8}{3 - 20 x})}$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (4 e^{\ln (\frac{15 x + 8}{3 - 20 x})}) + 15}$

Step 5.3.4.1.2

Factor $5$ out of $15$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (4 e^{\ln (\frac{15 x + 8}{3 - 20 x})}) + 5 (3)}$

Step 5.3.4.1.3

Factor $5$ out of $5 (4 e^{\ln (\frac{15 x + 8}{3 - 20 x})}) + 5 (3)$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (4 e^{\ln (\frac{15 x + 8}{3 - 20 x})} + 3)}$

Step 5.3.4.2

Exponentiation and log are inverse functions.

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (4 (\frac{15 x + 8}{3 - 20 x}) + 3)}$

Step 5.3.4.3

Combine $4$ and $\frac{15 x + 8}{3 - 20 x}$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (\frac{4 (15 x + 8)}{3 - 20 x} + 3)}$

Step 5.3.4.4

To write $3$ as a fraction with a common denominator, multiply by $\frac{3 - 20 x}{3 - 20 x}$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (\frac{4 (15 x + 8)}{3 - 20 x} + \frac{3 (3 - 20 x)}{3 - 20 x})}$

Step 5.3.4.5

Combine the numerators over the common denominator.

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (\frac{4 (15 x + 8) + 3 (3 - 20 x)}{3 - 20 x})}$

Step 5.3.4.6

Reorder terms.

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (\frac{4 (15 x + 8) + 3 (3 - 20 x)}{- 20 x + 3})}$

Step 5.3.4.7

Rewrite $\frac{4 (15 x + 8) + 3 (3 - 20 x)}{- 20 x + 3}$ in a factored form.

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Step 5.3.4.7.1

Apply the distributive property.

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (\frac{4 (15 x) + 4 \cdot 8 + 3 (3 - 20 x)}{- 20 x + 3})}$

Step 5.3.4.7.2

Multiply $15$ by $4$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (\frac{60 x + 4 \cdot 8 + 3 (3 - 20 x)}{- 20 x + 3})}$

Step 5.3.4.7.3

Multiply $4$ by $8$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (\frac{60 x + 32 + 3 (3 - 20 x)}{- 20 x + 3})}$

Step 5.3.4.7.4

Apply the distributive property.

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (\frac{60 x + 32 + 3 \cdot 3 + 3 (- 20 x)}{- 20 x + 3})}$

Step 5.3.4.7.5

Multiply $3$ by $3$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (\frac{60 x + 32 + 9 + 3 (- 20 x)}{- 20 x + 3})}$

Step 5.3.4.7.6

Multiply $- 20$ by $3$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (\frac{60 x + 32 + 9 - 60 x}{- 20 x + 3})}$

Step 5.3.4.7.7

Subtract $60 x$ from $60 x$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (\frac{0 + 32 + 9}{- 20 x + 3})}$

Step 5.3.4.7.8

Add $0$ and $32$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (\frac{32 + 9}{- 20 x + 3})}$

Step 5.3.4.7.9

Add $32$ and $9$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{5 (\frac{41}{- 20 x + 3})}$

Step 5.3.5

Combine fractions.

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Step 5.3.5.1

Combine $5$ and $\frac{41}{- 20 x + 3}$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{\frac{5 \cdot 41}{- 20 x + 3}}$

Step 5.3.5.2

Multiply $5$ by $41$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{\frac{205 x}{- 20 x + 3}}{\frac{205}{- 20 x + 3}}$

Step 5.3.6

Multiply the numerator by the reciprocal of the denominator.

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{205 x}{- 20 x + 3} \cdot \frac{- 20 x + 3}{205}$

Step 5.3.7

Cancel the common factor of $205$ .

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Step 5.3.7.1

Factor $205$ out of $205 x$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{205 (x)}{- 20 x + 3} \cdot \frac{- 20 x + 3}{205}$

Step 5.3.7.2

Cancel the common factor.

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{205 x}{- 20 x + 3} \cdot \frac{- 20 x + 3}{205}$

Step 5.3.7.3

Rewrite the expression.

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{x}{- 20 x + 3} \cdot (- 20 x + 3)$

Step 5.3.8

Multiply $\frac{x}{- 20 x + 3}$ by $- 20 x + 3$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{x (- 20 x + 3)}{- 20 x + 3}$

Step 5.3.9

Cancel the common factor of $- 20 x + 3$ .

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Step 5.3.9.1

Cancel the common factor.

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = \frac{x (- 20 x + 3)}{- 20 x + 3}$

Step 5.3.9.2

Divide $x$ by $1$ .

$f (\ln (\frac{15 x + 8}{3 - 20 x})) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \ln (\frac{15 x + 8}{3 - 20 x})$ is the inverse of $f (x) = \frac{3 e^{x} - 8}{20 e^{x} + 15}$ .

$f^{- 1} (x) = \ln (\frac{15 x + 8}{3 - 20 x})$