Algebra Examples

$g (x) = - 3 x$

Step 1

Write $g (x) = - 3 x$ as an equation.

$y = - 3 x$

Step 2

Interchange the variables.

$x = - 3 y$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- 3 y = x$ .

$- 3 y = x$

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 3$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide $y$ by $1$ .

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

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

Step 4

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

$g^{- 1} (x) = - \frac{x}{3}$

Step 5

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

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

$g^{- 1} (- 3 x) = - \frac{- 3 x}{3}$

Step 5.2.3

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

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Step 5.2.3.1

Factor $3$ out of $- 3 x$ .

$g^{- 1} (- 3 x) = - \frac{3 (- x)}{3}$

Step 5.2.3.2

Cancel the common factors.

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Step 5.2.3.2.1

Factor $3$ out of $3$ .

$g^{- 1} (- 3 x) = - \frac{3 (- x)}{3 (1)}$

Step 5.2.3.2.2

Cancel the common factor.

$g^{- 1} (- 3 x) = - \frac{3 (- x)}{3 \cdot 1}$

Step 5.2.3.2.3

Rewrite the expression.

$g^{- 1} (- 3 x) = - \frac{- x}{1}$

Step 5.2.3.2.4

Divide $- x$ by $1$ .

$g^{- 1} (- 3 x) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

$g (- \frac{x}{3}) = - 3 (- \frac{x}{3})$

Step 5.3.3

Cancel the common factor of $3$ .

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Step 5.3.3.1

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

$g (- \frac{x}{3}) = - 3 \frac{- x}{3}$

Step 5.3.3.2

Factor $3$ out of $- 3$ .

$g (- \frac{x}{3}) = 3 (- 1) (\frac{- x}{3})$

Step 5.3.3.3

Cancel the common factor.

$g (- \frac{x}{3}) = 3 \cdot (- 1 \frac{- x}{3})$

Step 5.3.3.4

Rewrite the expression.

$g (- \frac{x}{3}) = - 1 (- x)$

Step 5.3.4

Multiply.

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Step 5.3.4.1

Multiply $- 1$ by $- 1$ .

$g (- \frac{x}{3}) = 1 x$

Step 5.3.4.2

Multiply $x$ by $1$ .

$g (- \frac{x}{3}) = x$

Step 5.4

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

$g^{- 1} (x) = - \frac{x}{3}$