For \( f(x)=3 x-3 \) and \( g(x)=4 x^{2}-3 \), find the following functions. a. \( (f \circ g)(x) \); b. \( (g \circ f)(x) \); c. \( (f \circ g)(-2) \); d. \( (g \circ f)(-2) \) a. \( (f \circ g)(x)=\square \) (Simplify your answer.)

To find \( (f \circ g)(x) \), we need to substitute \( g(x) \) into \( f(x) \). 1. First, calculate \( g(x) \): \[ g(x) = 4x^2 - 3 \] 2. Now, substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(4x^2 - 3) = 3(4x^2 - 3) - 3 \] 3. Simplify: \[ = 12x^2 - 9 - 3 = 12x^2 - 12 \] So, \[ (f \circ g)(x) = 12x^2 - 12 \] For part c, we now find \( (f \circ g)(-2) \): 1. First, evaluate \( g(-2) \): \[ g(-2) = 4(-2)^2 - 3 = 4(4) - 3 = 16 - 3 = 13 \] 2. Now substitute this value into \( f(x) \): \[ f(g(-2)) = f(13) = 3(13) - 3 = 39 - 3 = 36 \] Thus, \[ (f \circ g)(-2) = 36 \] Now let's proceed to part b for \( (g \circ f)(x) \): 1. Substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(3x - 3) = 4(3x - 3)^2 - 3 \] 2. Expand \( (3x - 3)^2 \): \[ (3x - 3)^2 = 9x^2 - 18x + 9 \] So, \[ g(f(x)) = 4(9x^2 - 18x + 9) - 3 = 36x^2 - 72x + 36 - 3 \] Simplifying further gives: \[ = 36x^2 - 72x + 33 \] Therefore, \[ (g \circ f)(x) = 36x^2 - 72x + 33 \] Now for part d, we find \( (g \circ f)(-2) \): 1. First, evaluate \( f(-2) \): \[ f(-2) = 3(-2) - 3 = -6 - 3 = -9 \] 2. Now substitute this value into \( g(x) \): \[ g(f(-2)) = g(-9) = 4(-9)^2 - 3 = 4(81) - 3 = 324 - 3 = 321 \] Thus, \[ (g \circ f)(-2) = 321 \] Final answers: a. \( (f \circ g)(x) = 12x^2 - 12 \) c. \( (f \circ g)(-2) = 36 \) b. \( (g \circ f)(x) = 36x^2 - 72x + 33 \) d. \( (g \circ f)(-2) = 321 \)