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Answer:
Step-by-step explanation:
The domain is the set of x-values for which the function is applicable. The pieces of a piecewise-defined function are each defined on their own domain. Here, there are three different functions, each defined on a different domain.
From left to right, the function's domain can be divided into the sections ...
- x ≤ -1
- -1 < x < 3
- 3 ≤ x
This section of the graph looks like a parabola that opens downward. Its vertex is (-4, 3), and it seems to have a scale factor less than 1.
For some scale factor 'a', the function in vertex form is ...
y = a(x -h)² +k . . . . . . quadratic with vertex (h, k)
y = a(x +4)² +3
We know the point (x, y) = (-1, 0) is on the curve, so we can use these values to find 'a':
0 = a(-1 +4)² +3 = 9a +3
a = -3/9 = -1/3
So, the left-section function is ...
m(x) = -1/3(x +4)² +3
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<h3>-1 < x < 3</h3>The middle section of the graph has increasing slope, so might be a parabola or an exponential function. We note the average rate of change goes from -4 in the interval (-1, 0) to -2 in the interval (0, 1) to -1 in the interval (1, 2). The slope changing by a constant factor (1/2) in each unit interval is characteristic of an exponential function. That factor is the base of the exponent.
The actual values on the curve also decrease by a factor of 1/2 in each unit interval, which tells us the function has not been translated vertically. The y-intercept value of 4 at x=0 tells us the multiplier of the function:
m(x) = 4(1/2)^x
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<h3>3 ≤ x</h3>The funciton in this domain is a straight line. It has a "rise" of -1 unit for each "run" of 1 unit, so its slope is -1. If we extend the line left to the y-axis, we see that it has a y-intercept of 5. Its equation is ...
m(x) = -x +5
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Putting the pieces together into one function description, we have ...
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<em>Additional comment</em>
If the function in the middle section were quadratic, its average rate of change on adjacent equal intervals would form an <em>arithmetic</em> sequence. Because the sequence is <em>geometric</em>, we know it is an exponential function.