Answer:
(f + g) (-2) = 1
Step-by-step explanation:
First, it is important to determine the quadratic and absolute value functions in order to perform the given function operations.
Quadratic Function: f(x)
Given the vertex (h, k) occurring at point, (-1, -4), substitute the value of these coordinates into the vertex form:
f(x) = a(x - h)² + k
where:
(h, k) = vertex
a = determines whether the graph opens up or down, and makes the parent function wider or narrower.
h = determines how far left or right the parent function is translated.
k = determines how far up or down the parent function is translated.
f(x) = a(x - h)² + k
f(x) = a(x + 1)² - 4
Next, to solve for the value of a, we need another point from the graph. Let's use one of the x-intercepts, (-3, 0):
f(x) = a(x + 1)² - 4
0 = a(-3 + 1)² - 4
0 = a(-2)² - 4
0 + 4 = 4a - 4 + 4
4 = 4a
4/4 = 4a/4
1 = a
Therefore, the quadratic function in vertex form is: f(x) = (x + 1)² - 4
Absolute value function, g(x):
Given the vertex (h, k) occurring at point, (2, 0), substitute the value of these coordinates into the absolute value function:
g(x) = a |x - h | + k
The definitions for a and the vertex, (h, k) remains the same.
Choose another point on the graph, (3, 1) and substitute the values of its coordinates, along with the vertex into the absolute value function to solve for the value of a:
g(x) = a |x - 2 | + 0
1 = a |3 - 2|
1 = a | 1 |
1 = a
Therefore, the absolute value function is: g(x) = |x - 2|.
Now, we are ready to solve for (f + g)(-2) = f(-2) + g(-2)
We'll solve each functions separately, then combine the solutions later on.
f(-2):
f(-2) = (-2 + 1)² - 4
f(-2) = (-1)² - 4
f(-2) = 1 - 4
f(-2) = -3
g(-2):
g(-2) = |-2 - 2|
g(-2) = | -4 |
g(-2) = 4
(f + g)(-2) = f(-2) + g(-2) = -3 + 4 = 1
Therefore, the correct answer is 1.
