Trigonometry Examples

$2 (3 + 6 x) + 14 x - 8$

Step 1

Interchange the variables.

$x = 2 (3 + 6 y) + 14 y - 8$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $2 (3 + 6 y) + 14 y - 8 = x$ .

$2 (3 + 6 y) + 14 y - 8 = x$

Step 2.2

Simplify $2 (3 + 6 y) + 14 y - 8$ .

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Apply the distributive property.

$2 \cdot 3 + 2 (6 y) + 14 y - 8 = x$

Step 2.2.1.2

Multiply $2$ by $3$ .

$6 + 2 (6 y) + 14 y - 8 = x$

Step 2.2.1.3

Multiply $6$ by $2$ .

$6 + 12 y + 14 y - 8 = x$

Step 2.2.2

Simplify by adding terms.

Tap for more steps...

Step 2.2.2.1

Subtract $8$ from $6$ .

$12 y + 14 y - 2 = x$

Step 2.2.2.2

Add $12 y$ and $14 y$ .

$26 y - 2 = x$

Step 2.3

Add $2$ to both sides of the equation.

$26 y = x + 2$

Step 2.4

Divide each term in $26 y = x + 2$ by $26$ and simplify.

Tap for more steps...

Step 2.4.1

Divide each term in $26 y = x + 2$ by $26$ .

$\frac{26 y}{26} = \frac{x}{26} + \frac{2}{26}$

Step 2.4.2

Simplify the left side.

Tap for more steps...

Step 2.4.2.1

Cancel the common factor of $26$ .

Tap for more steps...

Step 2.4.2.1.1

Cancel the common factor.

$\frac{26 y}{26} = \frac{x}{26} + \frac{2}{26}$

Step 2.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{26} + \frac{2}{26}$

Step 2.4.3

Simplify the right side.

Tap for more steps...

Step 2.4.3.1

Cancel the common factor of $2$ and $26$ .

Tap for more steps...

Step 2.4.3.1.1

Factor $2$ out of $2$ .

$y = \frac{x}{26} + \frac{2 (1)}{26}$

Step 2.4.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.4.3.1.2.1

Factor $2$ out of $26$ .

$y = \frac{x}{26} + \frac{2 \cdot 1}{2 \cdot 13}$

Step 2.4.3.1.2.2

Cancel the common factor.

$y = \frac{x}{26} + \frac{2 \cdot 1}{2 \cdot 13}$

Step 2.4.3.1.2.3

Rewrite the expression.

$y = \frac{x}{26} + \frac{1}{13}$

Step 3

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

$f^{- 1} (x) = \frac{x}{26} + \frac{1}{13}$

Step 4

Verify if $f^{- 1} (x) = \frac{x}{26} + \frac{1}{13}$ is the inverse of $f (x) = 2 (3 + 6 x) + 14 x - 8$ .

Tap for more steps...

Step 4.1

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (2 (3 + 6 x) + 14 x - 8)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (2 (3 + 6 x) + 14 x - 8) = \frac{2 (3 + 6 x) + 14 x - 8}{26} + \frac{1}{13}$

Step 4.2.3

Simplify each term.

Tap for more steps...

Step 4.2.3.1

Cancel the common factor of $2 (3 + 6 x) + 14 x - 8$ and $26$ .

Tap for more steps...

Step 4.2.3.1.1

Factor $2$ out of $14 x$ .

$f^{- 1} (2 (3 + 6 x) + 14 x - 8) = \frac{2 (3 + 6 x) + 2 (7 x) - 8}{26} + \frac{1}{13}$

Step 4.2.3.1.2

Factor $2$ out of $2 (3 + 6 x) + 2 (7 x)$ .

$f^{- 1} (2 (3 + 6 x) + 14 x - 8) = \frac{2 (3 + 6 x + 7 x) - 8}{26} + \frac{1}{13}$

Step 4.2.3.1.3

Factor $2$ out of $- 8$ .

$f^{- 1} (2 (3 + 6 x) + 14 x - 8) = \frac{2 (3 + 6 x + 7 x) + 2 \cdot - 4}{26} + \frac{1}{13}$

Step 4.2.3.1.4

Factor $2$ out of $2 (3 + 6 x + 7 x) + 2 \cdot - 4$ .

$f^{- 1} (2 (3 + 6 x) + 14 x - 8) = \frac{2 (3 + 6 x + 7 x - 4)}{26} + \frac{1}{13}$

Step 4.2.3.1.5

Cancel the common factors.

Tap for more steps...

Step 4.2.3.1.5.1

Factor $2$ out of $26$ .

$f^{- 1} (2 (3 + 6 x) + 14 x - 8) = \frac{2 (3 + 6 x + 7 x - 4)}{2 (13)} + \frac{1}{13}$

Step 4.2.3.1.5.2

Cancel the common factor.

$f^{- 1} (2 (3 + 6 x) + 14 x - 8) = \frac{2 (3 + 6 x + 7 x - 4)}{2 \cdot 13} + \frac{1}{13}$

Step 4.2.3.1.5.3

Rewrite the expression.

$f^{- 1} (2 (3 + 6 x) + 14 x - 8) = \frac{3 + 6 x + 7 x - 4}{13} + \frac{1}{13}$

Step 4.2.3.2

Simplify the numerator.

Tap for more steps...

Step 4.2.3.2.1

Subtract $4$ from $3$ .

$f^{- 1} (2 (3 + 6 x) + 14 x - 8) = \frac{6 x + 7 x - 1}{13} + \frac{1}{13}$

Step 4.2.3.2.2

Add $6 x$ and $7 x$ .

$f^{- 1} (2 (3 + 6 x) + 14 x - 8) = \frac{13 x - 1}{13} + \frac{1}{13}$

Step 4.2.4

Simplify terms.

Tap for more steps...

Step 4.2.4.1

Combine the numerators over the common denominator.

$f^{- 1} (2 (3 + 6 x) + 14 x - 8) = \frac{13 x - 1 + 1}{13}$

Step 4.2.4.2

Combine the opposite terms in $13 x - 1 + 1$ .

Tap for more steps...

Step 4.2.4.2.1

Add $- 1$ and $1$ .

$f^{- 1} (2 (3 + 6 x) + 14 x - 8) = \frac{13 x + 0}{13}$

Step 4.2.4.2.2

Add $13 x$ and $0$ .

$f^{- 1} (2 (3 + 6 x) + 14 x - 8) = \frac{13 x}{13}$

Step 4.2.4.3

Cancel the common factor of $13$ .

Tap for more steps...

Step 4.2.4.3.1

Cancel the common factor.

$f^{- 1} (2 (3 + 6 x) + 14 x - 8) = \frac{13 x}{13}$

Step 4.2.4.3.2

Divide $x$ by $1$ .

$f^{- 1} (2 (3 + 6 x) + 14 x - 8) = x$

Step 4.3

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

Tap for more steps...

Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{x}{26} + \frac{1}{13})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{26} + \frac{1}{13}) = 2 (3 + 6 (\frac{x}{26} + \frac{1}{13})) + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3

Simplify each term.

Tap for more steps...

Step 4.3.3.1

Simplify each term.

Tap for more steps...

Step 4.3.3.1.1

Apply the distributive property.

$f (\frac{x}{26} + \frac{1}{13}) = 2 (3 + 6 (\frac{x}{26}) + 6 (\frac{1}{13})) + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.3.1.2.1

Factor $2$ out of $6$ .

$f (\frac{x}{26} + \frac{1}{13}) = 2 (3 + 2 (3) (\frac{x}{26}) + 6 (\frac{1}{13})) + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.1.2.2

Factor $2$ out of $26$ .

$f (\frac{x}{26} + \frac{1}{13}) = 2 (3 + 2 \cdot (3 (\frac{x}{2 \cdot 13})) + 6 (\frac{1}{13})) + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.1.2.3

Cancel the common factor.

$f (\frac{x}{26} + \frac{1}{13}) = 2 (3 + 2 \cdot (3 (\frac{x}{2 \cdot 13})) + 6 (\frac{1}{13})) + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.1.2.4

Rewrite the expression.

$f (\frac{x}{26} + \frac{1}{13}) = 2 (3 + 3 (\frac{x}{13}) + 6 (\frac{1}{13})) + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.1.3

Combine $3$ and $\frac{x}{13}$ .

$f (\frac{x}{26} + \frac{1}{13}) = 2 (3 + \frac{3 x}{13} + 6 (\frac{1}{13})) + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.1.4

Combine $6$ and $\frac{1}{13}$ .

$f (\frac{x}{26} + \frac{1}{13}) = 2 (3 + \frac{3 x}{13} + \frac{6}{13}) + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$f (\frac{x}{26} + \frac{1}{13}) = 2 (\frac{3 x}{13} + 3 \cdot \frac{13}{13} + \frac{6}{13}) + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.3

Combine $3$ and $\frac{13}{13}$ .

$f (\frac{x}{26} + \frac{1}{13}) = 2 (\frac{3 x}{13} + \frac{3 \cdot 13}{13} + \frac{6}{13}) + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.4

Combine the numerators over the common denominator.

$f (\frac{x}{26} + \frac{1}{13}) = 2 (\frac{3 x}{13} + \frac{3 \cdot 13 + 6}{13}) + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.5

Simplify the numerator.

Tap for more steps...

Step 4.3.3.5.1

Multiply $3$ by $13$ .

$f (\frac{x}{26} + \frac{1}{13}) = 2 (\frac{3 x}{13} + \frac{39 + 6}{13}) + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.5.2

Add $39$ and $6$ .

$f (\frac{x}{26} + \frac{1}{13}) = 2 (\frac{3 x}{13} + \frac{45}{13}) + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.6

Apply the distributive property.

$f (\frac{x}{26} + \frac{1}{13}) = 2 (\frac{3 x}{13}) + 2 (\frac{45}{13}) + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.7

Multiply $2 \frac{3 x}{13}$ .

Tap for more steps...

Step 4.3.3.7.1

Combine $2$ and $\frac{3 x}{13}$ .

$f (\frac{x}{26} + \frac{1}{13}) = \frac{2 (3 x)}{13} + 2 (\frac{45}{13}) + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.7.2

Multiply $3$ by $2$ .

$f (\frac{x}{26} + \frac{1}{13}) = \frac{6 x}{13} + 2 (\frac{45}{13}) + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.8

Multiply $2 (\frac{45}{13})$ .

Tap for more steps...

Step 4.3.3.8.1

Combine $2$ and $\frac{45}{13}$ .

$f (\frac{x}{26} + \frac{1}{13}) = \frac{6 x}{13} + \frac{2 \cdot 45}{13} + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.8.2

Multiply $2$ by $45$ .

$f (\frac{x}{26} + \frac{1}{13}) = \frac{6 x}{13} + \frac{90}{13} + 14 (\frac{x}{26} + \frac{1}{13}) - 8$

Step 4.3.3.9

Apply the distributive property.

$f (\frac{x}{26} + \frac{1}{13}) = \frac{6 x}{13} + \frac{90}{13} + 14 (\frac{x}{26}) + 14 (\frac{1}{13}) - 8$

Step 4.3.3.10

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.3.10.1

Factor $2$ out of $14$ .

$f (\frac{x}{26} + \frac{1}{13}) = \frac{6 x}{13} + \frac{90}{13} + 2 (7) (\frac{x}{26}) + 14 (\frac{1}{13}) - 8$

Step 4.3.3.10.2

Factor $2$ out of $26$ .

$f (\frac{x}{26} + \frac{1}{13}) = \frac{6 x}{13} + \frac{90}{13} + 2 \cdot (7 (\frac{x}{2 \cdot 13})) + 14 (\frac{1}{13}) - 8$

Step 4.3.3.10.3

Cancel the common factor.

$f (\frac{x}{26} + \frac{1}{13}) = \frac{6 x}{13} + \frac{90}{13} + 2 \cdot (7 (\frac{x}{2 \cdot 13})) + 14 (\frac{1}{13}) - 8$

Step 4.3.3.10.4

Rewrite the expression.

$f (\frac{x}{26} + \frac{1}{13}) = \frac{6 x}{13} + \frac{90}{13} + 7 (\frac{x}{13}) + 14 (\frac{1}{13}) - 8$

Step 4.3.3.11

Combine $7$ and $\frac{x}{13}$ .

$f (\frac{x}{26} + \frac{1}{13}) = \frac{6 x}{13} + \frac{90}{13} + \frac{7 x}{13} + 14 (\frac{1}{13}) - 8$

Step 4.3.3.12

Combine $14$ and $\frac{1}{13}$ .

$f (\frac{x}{26} + \frac{1}{13}) = \frac{6 x}{13} + \frac{90}{13} + \frac{7 x}{13} + \frac{14}{13} - 8$

Step 4.3.4

Simplify terms.

Tap for more steps...

Step 4.3.4.1

Combine the numerators over the common denominator.

$f (\frac{x}{26} + \frac{1}{13}) = - 8 + \frac{6 x + 90 + 7 x + 14}{13}$

Step 4.3.4.2

Add $6 x$ and $7 x$ .

$f (\frac{x}{26} + \frac{1}{13}) = - 8 + \frac{13 x + 90 + 14}{13}$

Step 4.3.4.3

Add $90$ and $14$ .

$f (\frac{x}{26} + \frac{1}{13}) = - 8 + \frac{13 x + 104}{13}$

Step 4.3.4.4

Cancel the common factor of $13 x + 104$ and $13$ .

Tap for more steps...

Step 4.3.4.4.1

Factor $13$ out of $13 x$ .

$f (\frac{x}{26} + \frac{1}{13}) = - 8 + \frac{13 (x) + 104}{13}$

Step 4.3.4.4.2

Factor $13$ out of $104$ .

$f (\frac{x}{26} + \frac{1}{13}) = - 8 + \frac{13 (x) + 13 \cdot 8}{13}$

Step 4.3.4.4.3

Factor $13$ out of $13 (x) + 13 (8)$ .

$f (\frac{x}{26} + \frac{1}{13}) = - 8 + \frac{13 (x + 8)}{13}$

Step 4.3.4.4.4

Cancel the common factors.

Tap for more steps...

Step 4.3.4.4.4.1

Factor $13$ out of $13$ .

$f (\frac{x}{26} + \frac{1}{13}) = - 8 + \frac{13 (x + 8)}{13 (1)}$

Step 4.3.4.4.4.2

Cancel the common factor.

$f (\frac{x}{26} + \frac{1}{13}) = - 8 + \frac{13 (x + 8)}{13 \cdot 1}$

Step 4.3.4.4.4.3

Rewrite the expression.

$f (\frac{x}{26} + \frac{1}{13}) = - 8 + \frac{x + 8}{1}$

Step 4.3.4.4.4.4

Divide $x + 8$ by $1$ .

$f (\frac{x}{26} + \frac{1}{13}) = - 8 + x + 8$

Step 4.3.4.5

Combine the opposite terms in $- 8 + x + 8$ .

Tap for more steps...

Step 4.3.4.5.1

Add $- 8$ and $8$ .

$f (\frac{x}{26} + \frac{1}{13}) = x + 0$

Step 4.3.4.5.2

Add $x$ and $0$ .

$f (\frac{x}{26} + \frac{1}{13}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{26} + \frac{1}{13}$ is the inverse of $f (x) = 2 (3 + 6 x) + 14 x - 8$ .

$f^{- 1} (x) = \frac{x}{26} + \frac{1}{13}$