Graph the following function on the axes provided. $$ f(x)=\left\{\begin{array}{lll}4 & \text { for } & x \leq-3 \ 3 x & \text { for } & x2\end{array}\right. $$

Question

Graph the following function on the axes provided. $$ f(x)=\left\{\begin{array}{lll}4 & \text { for } & x \leq-3 \ 3 x & \text { for } & x>2\end{array}\right. $$

UpStudy AI · Accepted Answer

1. Define the pieces of the function:

- For $$ x\leq -3 $$, the function is constant:  
     $$
     f(x)=4.
     $$
     
   - For $$ x>2 $$, the function is linear:  
     $$
     f(x)=3x.
     $$

2. Graph the first piece ($$ x\leq-3 $$):

- Draw a horizontal line at $$ y=4 $$ that extends to the left from $$ x=-3 $$ (including all $$ x\leq -3 $$).
   - At the endpoint $$ x=-3 $$, place a closed circle at the point $$(-3,4)$$ to indicate that the value is defined there.

3. Graph the second piece ($$ x>2 $$):

- The graph is the line for $$ f(x)=3x $$.
   - Note that the function is only defined for $$ x>2 $$. At $$ x=2 $$, compute $$ f(2)=3(2)=6 $$. Since $$ x=2 $$ is not included in the domain for this piece, draw an open circle at the point $$(2,6)$$.
   - Draw the line through the point $$(2,6)$$ for all $$ x>2 $$.

4. Note the gaps:

- There is no definition of $$ f(x) $$ when $$-3 < x \leq 2$$. This part of the graph remains blank.

5. Final sketch:

- The horizontal line at $$ y=4 $$ for $$ x\leq -3 $$ with the closed dot at $$(-3,4)$$.
   - The line $$ y=3x $$ for $$ x>2 $$, starting with an open circle at $$(2,6)$$ and continuing to the right.

This completes the step-by-step construction of the graph for the given piecewise function.
To graph the function $$ f(x) $$, draw a horizontal line at $$ y=4 $$ for all $$ x \leq -3 $$ with a closed circle at $$(-3,4)$$. For $$ x > 2 $$, draw the line $$ y=3x $$ starting with an open circle at $$(2,6)$$ and extending to the right. There is no graph between $$ x = -3 $$ and $$ x = 2 $$.

Graph the following function on the axes provided.

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