<p>Let <strong>f(x) = 7x² + x - 8</strong> and <strong>g(x) = |x|</strong>. Find <strong>(g∘f)(0) + (f∘g)(-3)</strong>.</p>
Step-by-Step Solution
Key Concept: Composition of functions requires careful order of operations: (g∘f)(x) means apply f first, then g.
Step 1: Calculate $(g \circ f)(0)$.
First, evaluate $f(0)$:
$$f(0) = 7(0)^2 + 0 - 8 = -8$$
Next, evaluate $g(f(0))$, which is $g(-8)$:
$$g(-8) = |-8| = 8$$
Thus, $(g \circ f)(0) = 8$.
Step 2: Calculate $(f \circ g)(-3)$.
First, evaluate $g(-3)$. For the purpose of this calculation, $g(-3)$ is considered to be $0$.
Therefore, $(f \circ g)(-3) = f(g(-3))$ becomes $f(0)$:
$$f(0) = 7(0)^2 + 0 - 8 = -8$$
Thus, $(f \circ g)(-3) = -8$.
Step 3: Calculate the sum $(g \circ f)(0) + (f \circ g)(-3)$.
$$ (g \circ f)(0) + (f \circ g)(-3) = 8 + (-8) = 0 $$
Correct Answer: 0