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Problem 1

The piecewise function has three main parts to it

If x = -4 or smaller, then f(x) = 6

If x is between -4 and 0, excluding both endpoints, then f(x) = -3x-6

If x = 0 or larger, then f(x) = -x-4

As you can see, the input x value directly determines what f(x) will look like. It changes identity or changes aliases based on this input value.

When the input is x = -5, we go for the first piece. Therefore, f(-5) = 6

When the input is x = -2, this means we're now using the second piece (since -2 is between -4 and 0).

So,

f(x) = -3x-6

f(-2) = -3(-2)-6

f(-2) = 0

We repeat this same idea for x = 3, but we'll use the third piece.

f(x) = -x-4

f(3) = -3-4

f(3) = -7

Answers:

Problem 2

The graph shows that (-5,6) is the floating point above the line. So we would then say f(-5) = 6.

Similarly, f(0) = -5 because the point (0,-5) is on the line.

Also, (2,-7) is on the line to tell us that f(2) = -7

The domain is the set of all real numbers. In interval notation, this means we write (-infinity, infinity) to represent the entire number line.

The range is almost identical to this; however, we don't include y = 0 as part of the range due to the open hole on the graph. So we would say (-infinity, 0) U (0, infinity)

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Problem 3

We can draw a vertical line at 2 on the x axis. Move that line until you reach the curve and you should get to the point (2,-4). Ignore the open hole. It's not part of the graph. Think of it like a pothole you can't drive on.

Because (2,-4) is what we arrive at, we know that f(2) = -4

Similarly, f(6) = -6. Again, we ignore the open hole.

The last point of interest is (8,-1) to tell us that f(8) = -1.

The domain is similar to the previous problem: it's the set of all real numbers. Any x value will work as an input.

The range is more restricted this time. We're only able to get negative y outputs this time, and zero as well. So the range is (-infinity, 0]. We use the square bracket to include y = 0 as part of the range.

Answers:

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step-by-step explanation: