Math

QuestionGraph the piecewise function: f(x)={3,<x1x3,1<x2x/2,2<xf(x)=\begin{cases} 3, & -\infty<x\leq-1 \\ x-3, & -1<x\leq2 \\ x/2, & 2<x\leq\infty \end{cases}

Studdy Solution

STEP 1

Assumptions1. The function is defined in three different intervals <x1-\infty<x\leq-1, 1<x-1<x\leq, and <x<x\leq\infty. . In each interval, the function has a different definition f(x)=3f(x)=3, f(x)=x3f(x)=x-3, and f(x)=x/f(x)=x/ respectively.

STEP 2

We will graph each piece of the function separately. Let's start with the first piece f(x)=f(x)= for <x1-\infty<x\leq-1. This is a constant function, so it's a horizontal line at y=y=. We graph this line for all xx less than or equal to 1-1.

STEP 3

Next, we graph the second piece of the function f(x)=x3f(x)=x-3 for 1<x2-1<x\leq2. This is a linear function with a slope of 11 and a y-intercept of 3-3. We graph this line for all xx greater than 1-1 and less than or equal to 22.

STEP 4

Finally, we graph the third piece of the function f(x)=x/2f(x)=x/2 for 2<x2<x\leq\infty. This is also a linear function, but with a slope of 1/21/2 and a y-intercept of 00. We graph this line for all xx greater than 22.

STEP 5

It's important to note that the points where the function changes definition, x=1x=-1 and x=2x=2, are included in the intervals of only one piece of the function. This means that at x=1x=-1, the function takes the value f(1)=3f(-1)=3, and at x=2x=2, the function takes the value f(2)=23=1f(2)=2-3=-1.

STEP 6

Now, we combine all three pieces of the function to create the complete graph of the function. We make sure to include the points (1,3)(-1,3) and (2,1)(2,-1), and to draw open circles at the points (1,2)(-1,-2) and (2,1)(2,1) to indicate that these points are not included in the graph of the function.
The graph of the function f(x)f(x) is now complete.

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