Trigonometry Examples

$f (x) = \frac{3 x - 4}{9}$

Step 1

Write $f (x) = \frac{3 x - 4}{9}$ as an equation.

$y = \frac{3 x - 4}{9}$

Step 2

Interchange the variables.

$x = \frac{3 y - 4}{9}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\frac{3 y - 4}{9} = x$ .

$\frac{3 y - 4}{9} = x$

Step 3.2

Multiply both sides by $9$ .

$\frac{3 y - 4}{9} \cdot 9 = x \cdot 9$

Step 3.3

Simplify.

Tap for more steps...

Step 3.3.1

Simplify the left side.

Tap for more steps...

Step 3.3.1.1

Cancel the common factor of $9$ .

Tap for more steps...

Step 3.3.1.1.1

Cancel the common factor.

$\frac{3 y - 4}{9} \cdot 9 = x \cdot 9$

Step 3.3.1.1.2

Rewrite the expression.

$3 y - 4 = x \cdot 9$

Step 3.3.2

Simplify the right side.

Tap for more steps...

Step 3.3.2.1

Move $9$ to the left of $x$ .

$3 y - 4 = 9 x$

Step 3.4

Solve for $y$ .

Tap for more steps...

Step 3.4.1

Add $4$ to both sides of the equation.

$3 y = 9 x + 4$

Step 3.4.2

Divide each term in $3 y = 9 x + 4$ by $3$ and simplify.

Tap for more steps...

Step 3.4.2.1

Divide each term in $3 y = 9 x + 4$ by $3$ .

$\frac{3 y}{3} = \frac{9 x}{3} + \frac{4}{3}$

Step 3.4.2.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{9 x}{3} + \frac{4}{3}$

Step 3.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{9 x}{3} + \frac{4}{3}$

Step 3.4.2.3

Simplify the right side.

Tap for more steps...

Step 3.4.2.3.1

Cancel the common factor of $9$ and $3$ .

Tap for more steps...

Step 3.4.2.3.1.1

Factor $3$ out of $9 x$ .

$y = \frac{3 (3 x)}{3} + \frac{4}{3}$

Step 3.4.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 3.4.2.3.1.2.1

Factor $3$ out of $3$ .

$y = \frac{3 (3 x)}{3 (1)} + \frac{4}{3}$

Step 3.4.2.3.1.2.2

Cancel the common factor.

$y = \frac{3 (3 x)}{3 \cdot 1} + \frac{4}{3}$

Step 3.4.2.3.1.2.3

Rewrite the expression.

$y = \frac{3 x}{1} + \frac{4}{3}$

Step 3.4.2.3.1.2.4

Divide $3 x$ by $1$ .

$y = 3 x + \frac{4}{3}$

Step 4

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

$f^{- 1} (x) = 3 x + \frac{4}{3}$

Step 5

Verify if $f^{- 1} (x) = 3 x + \frac{4}{3}$ is the inverse of $f (x) = \frac{3 x - 4}{9}$ .

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 $f^{- 1} (\frac{3 x - 4}{9})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{3 x - 4}{9}) = 3 (\frac{3 x - 4}{9}) + \frac{4}{3}$

Step 5.2.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.3.1

Factor $3$ out of $9$ .

$f^{- 1} (\frac{3 x - 4}{9}) = 3 (\frac{3 x - 4}{3 (3)}) + \frac{4}{3}$

Step 5.2.3.2

Cancel the common factor.

$f^{- 1} (\frac{3 x - 4}{9}) = 3 (\frac{3 x - 4}{3 \cdot 3}) + \frac{4}{3}$

Step 5.2.3.3

Rewrite the expression.

$f^{- 1} (\frac{3 x - 4}{9}) = \frac{3 x - 4}{3} + \frac{4}{3}$

Step 5.2.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{3 x - 4}{9}) = \frac{3 x - 4 + 4}{3}$

Step 5.2.5

Combine the opposite terms in $3 x - 4 + 4$ .

Tap for more steps...

Step 5.2.5.1

Add $- 4$ and $4$ .

$f^{- 1} (\frac{3 x - 4}{9}) = \frac{3 x + 0}{3}$

Step 5.2.5.2

Add $3 x$ and $0$ .

$f^{- 1} (\frac{3 x - 4}{9}) = \frac{3 x}{3}$

Step 5.2.6

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.6.1

Cancel the common factor.

$f^{- 1} (\frac{3 x - 4}{9}) = \frac{3 x}{3}$

Step 5.2.6.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{3 x - 4}{9}) = x$

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 $f (3 x + \frac{4}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (3 x + \frac{4}{3}) = \frac{3 (3 x + \frac{4}{3}) - 4}{9}$

Step 5.3.3

Simplify the numerator.

Tap for more steps...

Step 5.3.3.1

Apply the distributive property.

$f (3 x + \frac{4}{3}) = \frac{3 (3 x) + 3 (\frac{4}{3}) - 4}{9}$

Step 5.3.3.2

Multiply $3$ by $3$ .

$f (3 x + \frac{4}{3}) = \frac{9 x + 3 (\frac{4}{3}) - 4}{9}$

Step 5.3.3.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.3.3.3.1

Cancel the common factor.

$f (3 x + \frac{4}{3}) = \frac{9 x + 3 (\frac{4}{3}) - 4}{9}$

Step 5.3.3.3.2

Rewrite the expression.

$f (3 x + \frac{4}{3}) = \frac{9 x + 4 - 4}{9}$

Step 5.3.3.4

Subtract $4$ from $4$ .

$f (3 x + \frac{4}{3}) = \frac{9 x + 0}{9}$

Step 5.3.3.5

Add $9 x$ and $0$ .

$f (3 x + \frac{4}{3}) = \frac{9 x}{9}$

Step 5.3.4

Cancel the common factor of $9$ .

Tap for more steps...

Step 5.3.4.1

Cancel the common factor.

$f (3 x + \frac{4}{3}) = \frac{9 x}{9}$

Step 5.3.4.2

Divide $x$ by $1$ .

$f (3 x + \frac{4}{3}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 3 x + \frac{4}{3}$ is the inverse of $f (x) = \frac{3 x - 4}{9}$ .

$f^{- 1} (x) = 3 x + \frac{4}{3}$