For \( f(x)=5-x \) and \( g(x)=4 x^{2}+x+7 \), 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 the composed functions, we’ll start with \( (f \circ g)(x) \) and \( (g \circ f)(x) \). a. For \( (f \circ g)(x) \): \[ (f \circ g)(x) = f(g(x)) = f(4x^2 + x + 7) \] Substituting \( g(x) \) into \( f(x) \): \[ = 5 - (4x^2 + x + 7) = 5 - 4x^2 - x - 7 = -4x^2 - x - 2 \] So, \[ (f \circ g)(x) = -4x^2 - x - 2 \] b. For \( (g \circ f)(x) \): \[ (g \circ f)(x) = g(f(x)) = g(5 - x) \] Substituting \( f(x) \) into \( g(x) \): \[ = 4(5 - x)^2 + (5 - x) + 7 \] Now simplifying \( (5-x)^2 \): \[ = 25 - 10x + x^2 \] So, \[ g(f(x)) = 4(25 - 10x + x^2) + (5 - x) + 7 = 100 - 40x + 4x^2 + 5 - x + 7 = 4x^2 - 41x + 112 \] Thus, \[ (g \circ f)(x) = 4x^2 - 41x + 112 \] Now for the specific evaluations. c. Calculating \( (f \circ g)(2) \): \[ (f \circ g)(2) = -4(2^2) - 2 - 2 = -4(4) - 2 - 2 = -16 - 2 - 2 = -20 \] So, \[ (f \circ g)(2) = -20 \] d. Calculating \( (g \circ f)(2) \): \[ (g \circ f)(2) = 4(2)^2 - 41(2) + 112 = 4(4) - 82 + 112 = 16 - 82 + 112 = 46 \] So, \[ (g \circ f)(2) = 46 \] In summary: a. \( (f \circ g)(x) = -4x^2 - x - 2 \) b. \( (g \circ f)(x) = 4x^2 - 41x + 112 \) c. \( (f \circ g)(2) = -20 \) d. \( (g \circ f)(2) = 46 \)