Find the Inverse g(x)=-3x

$g (x) = - 3 x$

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 $y$ .

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

$- 3 y = x$

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

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

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

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

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 $g (x) = - 3 x$ .

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

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

Set up the composite result function.

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

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

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

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

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

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- x$ by $1$ .

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

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

Set up the composite result function.

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

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

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

Cancel the common factor of $3$ .

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

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

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply.

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

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

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

Find the Inverse g(x)=-3x

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