Calculus Examples

$f (x) = \frac{6}{5} \cdot \cos (x)$

Step 1

Write $f (x) = \frac{6}{5} \cdot \cos (x)$ as an equation.

$y = \frac{6}{5} \cdot \cos (x)$

Step 2

Interchange the variables.

$x = \frac{6}{5} \cdot \cos (y)$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\frac{6}{5} \cdot \cos (y) = x$ .

$\frac{6}{5} \cdot \cos (y) = x$

Step 3.2

Multiply both sides of the equation by $\frac{5}{6}$ .

$\frac{5}{6} (\frac{6}{5} \cos (y)) = \frac{5}{6} x$

Step 3.3

Simplify both sides of the equation.

Tap for more steps...

Step 3.3.1

Simplify the left side.

Tap for more steps...

Step 3.3.1.1

Simplify $\frac{5}{6} (\frac{6}{5} \cos (y))$ .

Tap for more steps...

Step 3.3.1.1.1

Combine $\frac{6}{5}$ and $\cos (y)$ .

$\frac{5}{6} \cdot \frac{6 \cos (y)}{5} = \frac{5}{6} x$

Step 3.3.1.1.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.3.1.1.2.1

Cancel the common factor.

$\frac{5}{6} \cdot \frac{6 \cos (y)}{5} = \frac{5}{6} x$

Step 3.3.1.1.2.2

Rewrite the expression.

$\frac{1}{6} (6 \cos (y)) = \frac{5}{6} x$

Step 3.3.1.1.3

Cancel the common factor of $6$ .

Tap for more steps...

Step 3.3.1.1.3.1

Factor $6$ out of $6 \cos (y)$ .

$\frac{1}{6} (6 \cos (y)) = \frac{5}{6} x$

Step 3.3.1.1.3.2

Cancel the common factor.

$\frac{1}{6} (6 \cos (y)) = \frac{5}{6} x$

Step 3.3.1.1.3.3

Rewrite the expression.

$\cos (y) = \frac{5}{6} x$

Step 3.3.2

Simplify the right side.

Tap for more steps...

Step 3.3.2.1

Combine $\frac{5}{6}$ and $x$ .

$\cos (y) = \frac{5 x}{6}$

Step 3.4

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

$y = \arccos (\frac{5 x}{6})$

Step 4

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

$f^{- 1} (x) = \arccos (\frac{5 x}{6})$

Step 5

Verify if $f^{- 1} (x) = \arccos (\frac{5 x}{6})$ is the inverse of $f (x) = \frac{6}{5} \cdot \cos (x)$ .

Tap for more steps...

Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate $f^{- 1} (f (x))$ .

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\frac{6}{5} \cdot \cos (x))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{6}{5} \cdot \cos (x)) = \arccos (\frac{5 (\frac{6}{5} \cdot \cos (x))}{6})$

Step 5.2.3

Simplify the numerator.

Tap for more steps...

Step 5.2.3.1

Combine $5$ and $\frac{6}{5}$ .

$f^{- 1} (\frac{6}{5} \cdot \cos (x)) = \arccos (\frac{\frac{5 \cdot 6}{5} \cdot \cos (x)}{6})$

Step 5.2.3.2

Combine $\frac{5 \cdot 6}{5}$ and $\cos (x)$ .

$f^{- 1} (\frac{6}{5} \cdot \cos (x)) = \arccos (\frac{\frac{5 \cdot (6 \cos (x))}{5}}{6})$

Step 5.2.4

Multiply $5$ by $6$ .

$f^{- 1} (\frac{6}{5} \cdot \cos (x)) = \arccos (\frac{\frac{30 \cos (x)}{5}}{6})$

Step 5.2.5

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.2.5.1

Reduce the expression $\frac{30 \cos (x)}{5}$ by cancelling the common factors.

Tap for more steps...

Step 5.2.5.1.1

Factor $5$ out of $30 \cos (x)$ .

$f^{- 1} (\frac{6}{5} \cdot \cos (x)) = \arccos (\frac{\frac{5 (6 \cos (x))}{5}}{6})$

Step 5.2.5.1.2

Factor $5$ out of $5$ .

$f^{- 1} (\frac{6}{5} \cdot \cos (x)) = \arccos (\frac{\frac{5 (6 \cos (x))}{5 (1)}}{6})$

Step 5.2.5.1.3

Cancel the common factor.

$f^{- 1} (\frac{6}{5} \cdot \cos (x)) = \arccos (\frac{\frac{5 (6 \cos (x))}{5 \cdot 1}}{6})$

Step 5.2.5.1.4

Rewrite the expression.

$f^{- 1} (\frac{6}{5} \cdot \cos (x)) = \arccos (\frac{\frac{6 \cos (x)}{1}}{6})$

Step 5.2.5.2

Divide $6 \cos (x)$ by $1$ .

$f^{- 1} (\frac{6}{5} \cdot \cos (x)) = \arccos (\frac{6 \cos (x)}{6})$

Step 5.2.6

Cancel the common factor of $6$ .

Tap for more steps...

Step 5.2.6.1

Cancel the common factor.

$f^{- 1} (\frac{6}{5} \cdot \cos (x)) = \arccos (\frac{6 \cos (x)}{6})$

Step 5.2.6.2

Divide $\cos (x)$ by $1$ .

$f^{- 1} (\frac{6}{5} \cdot \cos (x)) = \arccos (\cos (x))$

Step 5.3

Evaluate $f (f^{- 1} (x))$ .

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\arccos (\frac{5 x}{6}))$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\arccos (\frac{5 x}{6})) = \frac{6}{5} \cdot \cos (\arccos (\frac{5 x}{6}))$

Step 5.3.3

The functions cosine and arccosine are inverses.

$f (\arccos (\frac{5 x}{6})) = \frac{6}{5} \cdot \frac{5 x}{6}$

Step 5.3.4

Cancel the common factor of $6$ .

Tap for more steps...

Step 5.3.4.1

Cancel the common factor.

$f (\arccos (\frac{5 x}{6})) = \frac{6}{5} \cdot \frac{5 x}{6}$

Step 5.3.4.2

Rewrite the expression.

$f (\arccos (\frac{5 x}{6})) = \frac{1}{5} \cdot (5 x)$

Step 5.3.5

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.3.5.1

Factor $5$ out of $5 x$ .

$f (\arccos (\frac{5 x}{6})) = \frac{1}{5} \cdot (5 (x))$

Step 5.3.5.2

Cancel the common factor.

$f (\arccos (\frac{5 x}{6})) = \frac{1}{5} \cdot (5 x)$

Step 5.3.5.3

Rewrite the expression.

$f (\arccos (\frac{5 x}{6})) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \arccos (\frac{5 x}{6})$ is the inverse of $f (x) = \frac{6}{5} \cdot \cos (x)$ .

$f^{- 1} (x) = \arccos (\frac{5 x}{6})$