Trigonometry Examples

$x - 3 y = 16$

Step 1

Subtract $x$ from both sides of the equation.

$- 3 y = 16 - x$

Step 2

Divide each term in $- 3 y = 16 - x$ by $- 3$ and simplify.

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Step 2.1

Divide each term in $- 3 y = 16 - x$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{16}{- 3} + \frac{- x}{- 3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{16}{- 3} + \frac{- x}{- 3}$

Step 2.2.1.2

Divide $y$ by $1$ .

$y = \frac{16}{- 3} + \frac{- x}{- 3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{16}{3} + \frac{- x}{- 3}$

Step 2.3.1.2

Dividing two negative values results in a positive value.

$y = - \frac{16}{3} + \frac{x}{3}$

Step 3

Interchange the variables.

$x = - \frac{16}{3} + \frac{y}{3}$

Step 4

Solve for $y$ .

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Step 4.1

Rewrite the equation as $- \frac{16}{3} + \frac{y}{3} = x$ .

$- \frac{16}{3} + \frac{y}{3} = x$

Step 4.2

Add $\frac{16}{3}$ to both sides of the equation.

$\frac{y}{3} = x + \frac{16}{3}$

Step 4.3

Multiply both sides of the equation by $3$ .

$3 \frac{y}{3} = 3 (x + \frac{16}{3})$

Step 4.4

Simplify both sides of the equation.

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Cancel the common factor of $3$ .

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Step 4.4.1.1.1

Cancel the common factor.

$3 \frac{y}{3} = 3 (x + \frac{16}{3})$

Step 4.4.1.1.2

Rewrite the expression.

$y = 3 (x + \frac{16}{3})$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify $3 (x + \frac{16}{3})$ .

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Step 4.4.2.1.1

Apply the distributive property.

$y = 3 x + 3 (\frac{16}{3})$

Step 4.4.2.1.2

Cancel the common factor of $3$ .

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Step 4.4.2.1.2.1

Cancel the common factor.

$y = 3 x + 3 (\frac{16}{3})$

Step 4.4.2.1.2.2

Rewrite the expression.

$y = 3 x + 16$

Step 5

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

$f^{- 1} (x) = 3 x + 16$

Step 6

Verify if $f^{- 1} (x) = 3 x + 16$ is the inverse of $f (x) = - \frac{16}{3} + \frac{x}{3}$ .

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Step 6.1

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

Step 6.2

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

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Step 6.2.1

Set up the composite result function.

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

Step 6.2.2

Evaluate $f^{- 1} (- \frac{16}{3} + \frac{x}{3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{16}{3} + \frac{x}{3}) = 3 (- \frac{16}{3} + \frac{x}{3}) + 16$

Step 6.2.3

Simplify each term.

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Step 6.2.3.1

Apply the distributive property.

$f^{- 1} (- \frac{16}{3} + \frac{x}{3}) = 3 (- \frac{16}{3}) + 3 (\frac{x}{3}) + 16$

Step 6.2.3.2

Cancel the common factor of $3$ .

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Step 6.2.3.2.1

Move the leading negative in $- \frac{16}{3}$ into the numerator.

$f^{- 1} (- \frac{16}{3} + \frac{x}{3}) = 3 (\frac{- 16}{3}) + 3 (\frac{x}{3}) + 16$

Step 6.2.3.2.2

Cancel the common factor.

$f^{- 1} (- \frac{16}{3} + \frac{x}{3}) = 3 (\frac{- 16}{3}) + 3 (\frac{x}{3}) + 16$

Step 6.2.3.2.3

Rewrite the expression.

$f^{- 1} (- \frac{16}{3} + \frac{x}{3}) = - 16 + 3 (\frac{x}{3}) + 16$

Step 6.2.3.3

Cancel the common factor of $3$ .

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Step 6.2.3.3.1

Cancel the common factor.

$f^{- 1} (- \frac{16}{3} + \frac{x}{3}) = - 16 + 3 (\frac{x}{3}) + 16$

Step 6.2.3.3.2

Rewrite the expression.

$f^{- 1} (- \frac{16}{3} + \frac{x}{3}) = - 16 + x + 16$

Step 6.2.4

Combine the opposite terms in $- 16 + x + 16$ .

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Step 6.2.4.1

Add $- 16$ and $16$ .

$f^{- 1} (- \frac{16}{3} + \frac{x}{3}) = x + 0$

Step 6.2.4.2

Add $x$ and $0$ .

$f^{- 1} (- \frac{16}{3} + \frac{x}{3}) = x$

Step 6.3

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

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Step 6.3.1

Set up the composite result function.

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

Step 6.3.2

Evaluate $f (3 x + 16)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (3 x + 16) = - \frac{16}{3} + \frac{3 x + 16}{3}$

Step 6.3.3

Combine the numerators over the common denominator.

$f (3 x + 16) = \frac{- 16 + 3 x + 16}{3}$

Step 6.3.4

Combine the opposite terms in $- 16 + 3 x + 16$ .

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Step 6.3.4.1

Add $- 16$ and $16$ .

$f (3 x + 16) = \frac{3 x + 0}{3}$

Step 6.3.4.2

Add $3 x$ and $0$ .

$f (3 x + 16) = \frac{3 x}{3}$

Step 6.3.5

Cancel the common factor of $3$ .

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Step 6.3.5.1

Cancel the common factor.

$f (3 x + 16) = \frac{3 x}{3}$

Step 6.3.5.2

Divide $x$ by $1$ .

$f (3 x + 16) = x$

Step 6.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 3 x + 16$ is the inverse of $f (x) = - \frac{16}{3} + \frac{x}{3}$ .

$f^{- 1} (x) = 3 x + 16$