Answer: C.
Explanation:
The function graphed is a piece-wise function.
A piece-wise function is a function where more than one formula is used to define the output over different pieces of the domain.
First lets look at the graph an identify the parts of this function:
We see that the first part of the function is a parabola, and from left to right, it starts from infinity and ends with a hole at (2, 8).
A hole is a single point where the graph is not defined and is indicated by an open circle.
The next part of the function starts with a closed circle at (2,2) and decreases at a rate of -1 (-1/1) to negative infinity.
With the graph broken down. lets identify which function in the answer choices is the graphed one.
From the given answer choices we see that all the functions have x^2 + 4 and -x + 4.
How ever each answer choice has different restrictions which can be further known as limits (calculus).
A limit is the value that a function approaches as the input approaches some value.
Lets look at each answer choices restrictions and compare it to the graph:
- For answer choice A, x^2 + 4, it has a restriction where x must be less than or equal to two. In comparison to the graph we see that that function doesn't include two, hence its open circle (hole) at (2, 8). Meaning that x must be only less than two, as that part of the function is approaching two but never equaling two. This makes A. incorrect.
- For answer choice B, x^2 + 4, it has a restriction where x must be greater than or equal to two. This is completely incorrect when compared to the graph, as that part of the function is not greater than two nor does it equal two. This makes B. incorrect.
- For answer choice C, x^2 + 4, it has a restriction where x must be less than two. This is true with the given graph as that part of the function contains values that are all less than two and doesn't equal two. To make sure it is fully correct, we must check the second part of the function, -x + 4. That part of the function has a restriction where c must be greater than or equal to two. This is true with the given graph as that part of the function starts on a close circle at (2, 2), showing that two is included, and the x values continue to increase as the function goes to negative infinity.
- For answer choice D, x^2 + 4, it has a restriction where x must be greater than two. This incorrect as that part of the function is not greater than two. This makes D. incorrect.