Trigonometry Examples

$f (x) = \frac{1}{4} \cdot \sec (π x)$

Step 1

Write $f (x) = \frac{1}{4} \cdot \sec (π x)$ as an equation.

$y = \frac{1}{4} \cdot \sec (π x)$

Step 2

Interchange the variables.

$x = \frac{1}{4} \cdot \sec (π y)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{1}{4} \cdot \sec (π y) = x$ .

$\frac{1}{4} \cdot \sec (π y) = x$

Step 3.2

Multiply both sides of the equation by $4$ .

$4 (\frac{1}{4} \sec (π y)) = 4 x$

Step 3.3

Simplify the left side.

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Step 3.3.1

Simplify $4 (\frac{1}{4} \sec (π y))$ .

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Step 3.3.1.1

Combine $\frac{1}{4}$ and $\sec (π y)$ .

$4 \frac{\sec (π y)}{4} = 4 x$

Step 3.3.1.2

Cancel the common factor of $4$ .

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Step 3.3.1.2.1

Cancel the common factor.

$4 \frac{\sec (π y)}{4} = 4 x$

Step 3.3.1.2.2

Rewrite the expression.

$\sec (π y) = 4 x$

Step 3.4

Take the inverse secant of both sides of the equation to extract $y$ from inside the secant.

$π y = arcsec (4 x)$

Step 3.5

Divide each term in $π y = arcsec (4 x)$ by $π$ and simplify.

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Step 3.5.1

Divide each term in $π y = arcsec (4 x)$ by $π$ .

$\frac{π y}{π} = \frac{arcsec (4 x)}{π}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $π$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{π y}{π} = \frac{arcsec (4 x)}{π}$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{arcsec (4 x)}{π}$

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{arcsec (4 x)}{π}$ is the inverse of $f (x) = \frac{1}{4} \cdot \sec (π x)$ .

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

$f^{- 1} (\frac{1}{4} \cdot \sec (π x)) = \frac{arcsec (4 (\frac{1}{4} \cdot \sec (π x)))}{π}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Cancel the common factor of $4$ .

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Step 5.2.3.1.1

Cancel the common factor.

$f^{- 1} (\frac{1}{4} \cdot \sec (π x)) = \frac{arcsec (4 (\frac{1}{4}) \cdot \sec (π x))}{π}$

Step 5.2.3.1.2

Rewrite the expression.

$f^{- 1} (\frac{1}{4} \cdot \sec (π x)) = \frac{arcsec (1 \cdot \sec (π x))}{π}$

Step 5.2.3.2

Multiply $\sec (π x)$ by $1$ .

$f^{- 1} (\frac{1}{4} \cdot \sec (π x)) = \frac{arcsec (\sec (π x))}{π}$

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

$f (\frac{arcsec (4 x)}{π}) = \frac{1}{4} \cdot \sec (π (\frac{arcsec (4 x)}{π}))$

Step 5.3.3

Cancel the common factor of $π$ .

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Step 5.3.3.1

Cancel the common factor.

$f (\frac{arcsec (4 x)}{π}) = \frac{1}{4} \cdot \sec (π (\frac{arcsec (4 x)}{π}))$

Step 5.3.3.2

Rewrite the expression.

$f (\frac{arcsec (4 x)}{π}) = \frac{1}{4} \cdot \sec (arcsec (4 x))$

Step 5.3.4

The functions secant and arcsecant are inverses.

$f (\frac{arcsec (4 x)}{π}) = \frac{1}{4} \cdot (4 x)$

Step 5.3.5

Cancel the common factor of $4$ .

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Step 5.3.5.1

Factor $4$ out of $4 x$ .

$f (\frac{arcsec (4 x)}{π}) = \frac{1}{4} \cdot (4 (x))$

Step 5.3.5.2

Cancel the common factor.

$f (\frac{arcsec (4 x)}{π}) = \frac{1}{4} \cdot (4 x)$

Step 5.3.5.3

Rewrite the expression.

$f (\frac{arcsec (4 x)}{π}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{arcsec (4 x)}{π}$ is the inverse of $f (x) = \frac{1}{4} \cdot \sec (π x)$ .

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