Trigonometry Examples

$y = - \frac{10^{x}}{4}$

Step 1

Interchange the variables.

$x = - \frac{10^{y}}{4}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $- \frac{10^{y}}{4} = x$ .

$- \frac{10^{y}}{4} = x$

Step 2.2

Multiply both sides of the equation by $- 4$ .

$- 4 (- \frac{10^{y}}{4}) = - 4 x$

Step 2.3

Simplify the left side.

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Step 2.3.1

Simplify $- 4 (- \frac{10^{y}}{4})$ .

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Step 2.3.1.1

Cancel the common factor of $4$ .

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Step 2.3.1.1.1

Move the leading negative in $- \frac{10^{y}}{4}$ into the numerator.

$- 4 \frac{- 10^{y}}{4} = - 4 x$

Step 2.3.1.1.2

Factor $4$ out of $- 4$ .

$4 (- 1) \frac{- 10^{y}}{4} = - 4 x$

Step 2.3.1.1.3

Cancel the common factor.

$4 \cdot - 1 \frac{- 10^{y}}{4} = - 4 x$

Step 2.3.1.1.4

Rewrite the expression.

$- - 10^{y} = - 4 x$

Step 2.3.1.2

Multiply.

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Step 2.3.1.2.1

Multiply $- 1$ by $- 1$ .

$1 \cdot 10^{y} = - 4 x$

Step 2.3.1.2.2

Multiply $10^{y}$ by $1$ .

$10^{y} = - 4 x$

Step 2.4

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{y}) = \log (- 4 x)$

Step 2.5

Expand the left side.

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Step 2.5.1

Expand $\log (10^{y})$ by moving $y$ outside the logarithm.

$y \log (10) = \log (- 4 x)$

Step 2.5.2

Logarithm base $10$ of $10$ is $1$ .

$y \cdot 1 = \log (- 4 x)$

Step 2.5.3

Multiply $y$ by $1$ .

$y = \log (- 4 x)$

Step 3

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

$f^{- 1} (x) = \log (- 4 x)$

Step 4

Verify if $f^{- 1} (x) = \log (- 4 x)$ is the inverse of $f (x) = - \frac{10^{x}}{4}$ .

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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 $f^{- 1} (- \frac{10^{x}}{4})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{10^{x}}{4}) = \log (- 4 (- \frac{10^{x}}{4}))$

Step 4.2.3

Cancel the common factor of $4$ .

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Step 4.2.3.1

Move the leading negative in $- \frac{10^{x}}{4}$ into the numerator.

$f^{- 1} (- \frac{10^{x}}{4}) = \log (- 4 \frac{- 10^{x}}{4})$

Step 4.2.3.2

Factor $4$ out of $- 4$ .

$f^{- 1} (- \frac{10^{x}}{4}) = \log (4 (- 1) (\frac{- 10^{x}}{4}))$

Step 4.2.3.3

Cancel the common factor.

$f^{- 1} (- \frac{10^{x}}{4}) = \log (4 \cdot (- 1 \frac{- 10^{x}}{4}))$

Step 4.2.3.4

Rewrite the expression.

$f^{- 1} (- \frac{10^{x}}{4}) = \log (- 1 (- 10^{x}))$

Step 4.2.4

Multiply.

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Step 4.2.4.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- \frac{10^{x}}{4}) = \log (1 \cdot 10^{x})$

Step 4.2.4.2

Multiply $10^{x}$ by $1$ .

$f^{- 1} (- \frac{10^{x}}{4}) = \log (10^{x})$

Step 4.2.5

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

$f^{- 1} (- \frac{10^{x}}{4}) = x \log (10)$

Step 4.2.6

Logarithm base $10$ of $10$ is $1$ .

$f^{- 1} (- \frac{10^{x}}{4}) = x \cdot 1$

Step 4.2.7

Multiply $x$ by $1$ .

$f^{- 1} (- \frac{10^{x}}{4}) = x$

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

$f (\log (- 4 x)) = - \frac{10^{\log (- 4 x)}}{4}$

Step 4.3.3

Exponentiation and log are inverse functions.

$f (\log (- 4 x)) = - \frac{- 4 x}{4}$

Step 4.3.4

Cancel the common factor of $- 4$ and $4$ .

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Step 4.3.4.1

Factor $4$ out of $- 4 x$ .

$f (\log (- 4 x)) = - \frac{4 (- x)}{4}$

Step 4.3.4.2

Cancel the common factors.

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Step 4.3.4.2.1

Factor $4$ out of $4$ .

$f (\log (- 4 x)) = - \frac{4 (- x)}{4 (1)}$

Step 4.3.4.2.2

Cancel the common factor.

$f (\log (- 4 x)) = - \frac{4 (- x)}{4 \cdot 1}$

Step 4.3.4.2.3

Rewrite the expression.

$f (\log (- 4 x)) = - \frac{- x}{1}$

Step 4.3.4.2.4

Divide $- x$ by $1$ .

$f (\log (- 4 x)) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \log (- 4 x)$ is the inverse of $f (x) = - \frac{10^{x}}{4}$ .

$f^{- 1} (x) = \log (- 4 x)$