Evaluate $g(x)=3 x$ when $x=-2, 0$, and $5$.

We are given the linear function $g(x) = 3x$. Our task is to determine the values of this function for three specific inputs: $x = -2$, $x = 0$, and $x = 5$. Evaluating a function at a point involves substituting that point's coordinate into the expression for $x$ and calculating the result.

Evaluating for $x = -2$

To find $g(-2)$, we substitute $-2$ into the equation:
\[g(-2) = 3 \times (-2) = -6.\]
This means that when the input is $-2$, the output of the function is $-6$.

Evaluating for $x = 0$

To find $g(0)$, we substitute $0$ into the equation:
\[g(0) = 3 \times 0 = 0.\]
This means that when the input is $0$, the output of the function is $0$.

Evaluating for $x = 5$

To find $g(5)$, we substitute $5$ into the equation:
\[g(5) = 3 \times 5 = 15.\]
This means that when the input is $5$, the output of the function is $15$.

Conclusion

We have successfully evaluated the function $g(x) = 3x$ at the requested points:

For $x = -2$, $g(-2) = -6$.
For $x = 0$, $g(0) = 0$.
For $x = 5$, $g(5) = 15$.

The calculated values are $\boxed{g(-2)=-6, g(0)=0, g(5)=15}$.

Examples

Linear functions represent proportional relationships where one quantity changes at a constant rate relative to another. For example, if a machine produces 3 parts per hour, the number of parts produced $g(x)$ after $x$ hours is $g(x) = 3x$. Using the evaluations above, we can determine how many parts are produced in specific time frames: 0 parts in 0 hours, 15 parts in 5 hours, or even interpret negative time as a look back in the past where, hypothetically, 6 parts would have been produced 2 hours ago.