\( 1 \leftarrow \) 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. \( (f \circ g)(x)=12 x^{2}-12 \) (Simplify your answer.) b. \( (g \circ f)(x)= \) (Simplify your answer.)

Simplify the expression by following steps: - step0: Subtract the terms: \(3\left(4x^{2}-3\right)-3\) - step1: Expand the expression: \(12x^{2}-9-3\) - step2: Subtract the numbers: \(12x^{2}-12\) Expand the expression \( 4(3x-3)^2-3 \) Simplify the expression by following steps: - step0: Subtract the terms: \(4\left(3x-3\right)^{2}-3\) - step1: Expand the expression: \(36x^{2}-72x+36-3\) - step2: Subtract the numbers: \(36x^{2}-72x+33\) Let's solve the problem step by step. ### Given Functions - \( f(x) = 3x - 3 \) - \( g(x) = 4x^2 - 3 \) ### a. Find \( (f \circ g)(x) \) This means we need to substitute \( g(x) \) into \( f(x) \): \[ (f \circ g)(x) = f(g(x)) = f(4x^2 - 3) \] Substituting \( g(x) \) into \( f(x) \): \[ f(4x^2 - 3) = 3(4x^2 - 3) - 3 \] Now, let's simplify: \[ = 12x^2 - 9 - 3 = 12x^2 - 12 \] Thus, \[ (f \circ g)(x) = 12x^2 - 12 \] ### b. Find \( (g \circ f)(x) \) This means we need to substitute \( f(x) \) into \( g(x) \): \[ (g \circ f)(x) = g(f(x)) = g(3x - 3) \] Substituting \( f(x) \) into \( g(x) \): \[ g(3x - 3) = 4(3x - 3)^2 - 3 \] Now, let's simplify: \[ = 4(9x^2 - 18x + 9) - 3 = 36x^2 - 72x + 36 - 3 = 36x^2 - 72x + 33 \] Thus, \[ (g \circ f)(x) = 36x^2 - 72x + 33 \] ### c. Find \( (f \circ g)(-2) \) Now we will evaluate \( (f \circ g)(x) \) at \( x = -2 \): \[ (f \circ g)(-2) = 12(-2)^2 - 12 = 12(4) - 12 = 48 - 12 = 36 \] ### d. Find \( (g \circ f)(-2) \) Now we will evaluate \( (g \circ f)(x) \) at \( x = -2 \): \[ (g \circ f)(-2) = 36(-2)^2 - 72(-2) + 33 = 36(4) + 144 + 33 = 144 + 144 + 33 = 321 \] ### Summary of Results - a. \( (f \circ g)(x) = 12x^2 - 12 \) - b. \( (g \circ f)(x) = 36x^2 - 72x + 33 \) - c. \( (f \circ g)(-2) = 36 \) - d. \( (g \circ f)(-2) = 321 \)