Trigonometry Examples

$f (x) = 1 + \sec (x)$

Step 1

Write $f (x) = 1 + \sec (x)$ as an equation.

$y = 1 + \sec (x)$

Step 2

Interchange the variables.

$x = 1 + \sec (y)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $1 + \sec (y) = x$ .

$1 + \sec (y) = x$

Step 3.2

Subtract $1$ from both sides of the equation.

$\sec (y) = x - 1$

Step 3.3

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

$y = arcsec (x - 1)$

Step 4

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

$f^{- 1} (x) = arcsec (x - 1)$

Step 5

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

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

$f^{- 1} (1 + \sec (x)) = arcsec ((1 + \sec (x)) - 1)$

Step 5.2.3

Combine the opposite terms in $(1 + \sec (x)) - 1$ .

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Step 5.2.3.1

Subtract $1$ from $1$ .

$f^{- 1} (1 + \sec (x)) = arcsec (0 + \sec (x))$

Step 5.2.3.2

Add $0$ and $\sec (x)$ .

$f^{- 1} (1 + \sec (x)) = arcsec (\sec (x))$

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

$f (arcsec (x - 1)) = 1 + \sec (arcsec (x - 1))$

Step 5.3.3

The functions secant and arcsecant are inverses.

$f (arcsec (x - 1)) = 1 + x - 1$

Step 5.3.4

Combine the opposite terms in $1 + x - 1$ .

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Step 5.3.4.1

Subtract $1$ from $1$ .

$f (arcsec (x - 1)) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (arcsec (x - 1)) = x$

Step 5.4

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

$f^{- 1} (x) = arcsec (x - 1)$