For \( f(x)=4-x \) and \( g(x)=4 x^{2}+x+6 \), find the following functions. \( \begin{array}{l}\text { a. }(f \circ g)(x) ; b .(g \circ f)(x) ; \text { c. }(f \circ g)(3) ; d .(g \circ f)(3) \\ \begin{array}{l}\text { a. }(f \circ g)(x)=\square \\ \text { (Simplify your answer. })\end{array}\end{array} \)

To find the desired functions, let's start by calculating \( (f \circ g)(x) \) and \( (g \circ f)(x) \): ### a. \( (f \circ g)(x) \) This means we need to substitute \( g(x) \) into \( f(x) \): \[ g(x) = 4x^2 + x + 6 \] Now we substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(4x^2 + x + 6) = 4 - (4x^2 + x + 6) \] Simplifying this: \[ f(g(x)) = 4 - 4x^2 - x - 6 = -4x^2 - x - 2 \] Therefore, \[ (f \circ g)(x) = -4x^2 - x - 2 \] ### b. \( (g \circ f)(x) \) This means we substitute \( f(x) \) into \( g(x) \): \[ f(x) = 4 - x \] Substituting into \( g(x) \): \[ g(f(x)) = g(4 - x) = 4(4 - x)^2 + (4 - x) + 6 \] Calculating \( (4 - x)^2 \): \[ (4-x)^2 = 16 - 8x + x^2 \] Thus: \[ g(f(x)) = 4(16 - 8x + x^2) + (4 - x) + 6 = 64 - 32x + 4x^2 + 4 - x + 6 \] Now combining like terms: \[ g(f(x)) = 4x^2 - 33x + 74 \] Therefore, \[ (g \circ f)(x) = 4x^2 - 33x + 74 \] ### c. \( (f \circ g)(3) \) Now, substitute \( x = 3 \) in \( (f \circ g)(x) \): \[ (f \circ g)(3) = -4(3)^2 - (3) - 2 = -4 \cdot 9 - 3 - 2 = -36 - 3 - 2 = -41 \] ### d. \( (g \circ f)(3) \) Substituting \( x = 3 \) in \( (g \circ f)(x) \): \[ (g \circ f)(3) = 4(3)^2 - 33(3) + 74 = 4 \cdot 9 - 99 + 74 = 36 - 99 + 74 = 11 \] In summary: - \( (f \circ g)(x) = -4x^2 - x - 2 \) - \( (g \circ f)(x) = 4x^2 - 33x + 74 \) - \( (f \circ g)(3) = -41 \) - \( (g \circ f)(3) = 11 \)