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Consider the function.

FU(X) = -x-24
Which conclusions can be drawn about f'(x)? Select two options.
a)F"(x) has a slope of __
b)f'(x) has a restricted domain.
c)f'(x) has a y-intercept of (0, -36).
d)f'(x) has an x-intercept of (-36,0).
e)f(x) has a range of all real numbers,

User Jwdink
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1 Answer

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Final answer:

The derivative of the function FU(x) = -x - 24 is a horizontal line with a constant slope of -1, indicating that it does not have a restricted domain or any intercepts with the x or y-axis. The original function has a range of all real numbers.

Step-by-step explanation:

When examining the function FU(x) = -x - 24, we can determine certain characteristics about its derivative, f'(x). First, to find f'(x), we differentiate FU(x), giving us f'(x) = -1. This tells us that the slope of the derivative's graph is constant and negative, represented by a horizontal line with a value of -1. There are no restrictions on the domain for a linear function such as this, so it does not have a restricted domain. Therefore, options (a) and (b) are not correct.

Since f'(x) = -1 is a horizontal line, it does not have a y-intercept of (0, -36) or any x-intercept because it never crosses the y-axis or x-axis; these being (c) and (d) are incorrect as well. Instead, the line is below the x-axis by 1 unit at all points along the x-axis. Lastly, for e), since the original function FU(x) is linear and goes to negative infinity as x goes to positive infinity, and vice versa, its range is indeed all real numbers. This confirms that option (e) is correct.

User Hughjdavey
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