Questions: Given f(x)=5x^2-4 and g(x)=4-(1/2)x^2, find the following expressions. (a) (f ∘ g)(4) (b) (g ∘ f)(2) (c) (f ∘ f)(1) (d) (g ∘ g)(0) (a) (f ∘ g)(4)= (Simplify your answer)

Transcript text: Given $f(x)=5 x^{2}-4$ and $g(x)=4-\frac{1}{2} x^{2}$, find the following expressions. (a) $(f \circ g)(4)$ (b) $(g \circ f)(2)$ (c) $(f \circ f)(1)$ (d) $(g \circ g)(0)$ (a) $(f \circ g)(4)=$ $\square$ (Simplify your answer)

Solution

Solution Steps

To solve these problems, we need to evaluate composite functions. For each part, substitute the given value into the inner function, then use the result as the input for the outer function.

(a) For $(f \circ g)(4)$, first evaluate $g(4)$, then substitute that result into $f(x)$.

(b) For $(g \circ f)(2)$, first evaluate $f(2)$, then substitute that result into $g(x)$.

Step 1: Calculate $ (f \circ g)(4) $

First, we evaluate $ g(4) $: \[ g(4) = 4 - \frac{1}{2} \cdot 4^2 = 4 - \frac{1}{2} \cdot 16 = 4 - 8 = -4 \] Next, we substitute this result into $ f(x) $: \[ f(g(4)) = f(-4) = 5 \cdot (-4)^2 - 4 = 5 \cdot 16 - 4 = 80 - 4 = 76 \] Thus, $ (f \circ g)(4) = 76 $.

Step 2: Calculate $ (g \circ f)(2) $

First, we evaluate $ f(2) $: \[ f(2) = 5 \cdot 2^2 - 4 = 5 \cdot 4 - 4 = 20 - 4 = 16 \] Next, we substitute this result into $ g(x) $: \[ g(f(2)) = g(16) = 4 - \frac{1}{2} \cdot 16^2 = 4 - \frac{1}{2} \cdot 256 = 4 - 128 = -124 \] Thus, $ (g \circ f)(2) = -124 $.

Step 3: Calculate $ (f \circ f)(1) $

First, we evaluate $ f(1) $: \[ f(1) = 5 \cdot 1^2 - 4 = 5 \cdot 1 - 4 = 5 - 4 = 1 \] Next, we substitute this result back into $ f(x) $: \[ f(f(1)) = f(1) = 1 \] Thus, $ (f \circ f)(1) = 1 $.