Answer: Choice A) As x approaches positive infinity, f(x) approaches positive infinity.
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Step-by-step explanation:
The graph is shown below. It's composed of the three functions
- g(x) = 2^x
- h(x) = -x^2 - 4x + 1
- j(x) = (1/2)x + 3
But we only graph a piece of each function as described in the instructions. We only graph g(x) when x is smaller than 0. We graph h(x) when 0 < x < 2. If x > 2, then we graph j(x)
As the graph indicates, the function is not increasing over the entire domain. The portion from x = 0 to x = 2 is decreasing, or going downhill, as we move from left to right. This is the blue portion of the piecewise function. Note how h(x) is decreasing when x > -2. So we can rule out choice B.
We can also rule out choice C. We have a disconnect in the curve and have two separate pieces. The green portion is not connected to the blue portion. So this is why the function is not continuous.
A nonvisual way to verify this is to note that h(2) = -11 while j(2) = 4. In order for the piecewise function to be continuous, we must have h(2) and j(2) be the same value.
Choice D is false as well because x = 0 and x = 2 are not part of the domain. When we say x < 0, we aren't including x = 0. We would need the "or equal to" part of the inequality sign. Similarly this applies to 0 < x < 2 as well. As you can see, x = 2 is also left out. So we're missing two values to form the entire set of real numbers.
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Choice A is true because as x gets larger, the j(x) function goes off forever upward. So overall, f(x) will move in that same direction. In other words, as you move to the right toward infinity, the graph goes upward forever.