Question

graph the piecewise function. f(x)= {3x-5 if x is less than or equal to -1. -2x+3 if -1 is less than x is less than 4. 2 if x is greater than of equal to 4.

Answer 1

Here's how to graph the piecewise function:

First, we graph the function for the first interval, which is f(x) = 3x - 5 when x ≤ -1. This is a straight line with a slope of 3 and a y-intercept of -5. Since this interval includes -1, we draw a closed circle at x = -1 to indicate that it is included in the interval. The line is decreasing as x increases.

Next, we graph the function for the second interval, which is f(x) = -2x + 3 when -1 < x < 4. This is also a straight line, but with a slope of -2 and a y-intercept of 3. Since this interval does not include -1, we draw an open circle at x = -1 to indicate that it is not included in the interval. We also draw an open circle at x = 4 to indicate that it is not included in the interval. The line is increasing as x increases.

Finally, we graph the function for the third interval, which is f(x) = 2 when x ≥ 4. This is a horizontal line at y = 2. Since this interval includes 4, we draw a closed circle at x = 4 to indicate that it is included in the interval.

When we put all three intervals together, we get a graph that looks like this:

```

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2 | /

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_______|_____________

-1 4

```

The graph consists of a downward-sloping line from (-∞, -1], an upward-sloping line from (-1, 4), and a horizontal line from [4, ∞).

Answer 2

To graph the piecewise function, we need to graph each piece of the function separately and then combine them.

First, let's graph the function f(x) = 3x - 5 for x ≤ -1. This is a line with slope 3 and y-intercept -5 that passes through the point (-1,-8) since the inequality includes the endpoint.

Next, let's graph the function f(x) = -2x + 3 for -1 < x < 4. This is a line with slope -2 and y-intercept 3 that passes through the point (-1,5) and (4,-5) since the inequality does not include the endpoints.

Finally, let's graph the function f(x) = 2 for x ≥ 4. This is a horizontal line at y = 2.

Now we can combine the three graphs to get the graph of the piecewise function. The graph consists of three line segments: a line segment with slope 3 from negative infinity to -1, a line segment with slope -2 from -1 to 4, and a horizontal line at y = 2 from 4 to infinity.

I hope this helps!