Questions: Question 8 For the following exercise, find the indicated function given f(x)=3x^2+2 and g(x)=2x-4. a. f(g(3))= 14 b. f(g(x))=12x^2-48x+50 c. g(f(x))=6x^2 d. (g ∘ g)(x)= e. (f ∘ f)(-2)=

Transcript text: Question 8 For the following exercise, find the indicated function given $f(x)=3 x^{2}+2$ and $g(x)=2 x-4$. a. $f(g(3))=$ 14 $\square$ b. $f(g(x))=12 x^{2}-48 x+50$ c. $g(f(x))=6 x^{2}$ d. $(g \circ g)(x)=$ $\square$ e. $(f \circ f)(-2)=$ $\square$

Solution

Solution Steps

Solution Approach

a. To find $ f(g(3)) $, first evaluate $ g(3) $ using the function $ g(x) = 2x - 4 $. Then, substitute the result into $ f(x) = 3x^2 + 2 $ to find $ f(g(3)) $.

b. To find $ f(g(x)) $, substitute $ g(x) = 2x - 4 $ into $ f(x) = 3x^2 + 2 $. Simplify the resulting expression to get $ f(g(x)) $.

c. To find $ g(f(x)) $, substitute $ f(x) = 3x^2 + 2 $ into $ g(x) = 2x - 4 $. Simplify the resulting expression to get $ g(f(x)) $.

Step 1: Calculate $ g(3) $

To find $ g(3) $, we use the function $ g(x) = 2x - 4 $: \[ g(3) = 2(3) - 4 = 6 - 4 = 2 \]

Step 2: Calculate $ f(g(3)) $

Now, we substitute $ g(3) $ into $ f(x) = 3x^2 + 2 $: \[ f(g(3)) = f(2) = 3(2^2) + 2 = 3(4) + 2 = 12 + 2 = 14 \]

Step 3: Calculate $ f(g(x)) $

Next, we substitute $ g(x) = 2x - 4 $ into $ f(x) $: \[ f(g(x)) = f(2x - 4) = 3(2x - 4)^2 + 2 \] Expanding $ (2x - 4)^2 $: \[ (2x - 4)^2 = 4x^2 - 16x + 16 \] Thus, \[ f(g(x)) = 3(4x^2 - 16x + 16) + 2 = 12x^2 - 48x + 48 + 2 = 12x^2 - 48x + 50 \]

Step 4: Calculate $ g(f(x)) $

Now, we substitute $ f(x) = 3x^2 + 2 $ into $ g(x) $: \[ g(f(x)) = g(3x^2 + 2) = 2(3x^2 + 2) - 4 = 6x^2 + 4 - 4 = 6x^2 \]