For all values of \( x, f(x)=2 x-3 \) and \( g(x)=x^{2}+2 \). Solve \( f g(x)=g f(x) \) by finding the values of \( x \). Select one: a. \( \quad x=0 \) b. \( \quad x=-5,-1 \) c. \( \quad x=5,1 \) d. \( \quad x=-5,1 \)

To solve the equation \( f(g(x)) = g(f(x)) \), we first find the expressions for \( f(g(x)) \) and \( g(f(x)) \). 1. Calculate \( g(x) \): \[ g(x) = x^2 + 2 \] Now apply \( f \): \[ f(g(x)) = f(x^2 + 2) = 2(x^2 + 2) - 3 = 2x^2 + 4 - 3 = 2x^2 + 1 \] 2. Next, calculate \( f(x) \): \[ f(x) = 2x - 3 \] Now apply \( g \): \[ g(f(x)) = g(2x - 3) = (2x - 3)^2 + 2 \] Expanding it gives: \[ (2x - 3)^2 = 4x^2 - 12x + 9 \] Therefore, \[ g(f(x)) = 4x^2 - 12x + 9 + 2 = 4x^2 - 12x + 11 \] Now we set the two expressions equal: \[ 2x^2 + 1 = 4x^2 - 12x + 11 \] Rearranging gives: \[ 0 = 4x^2 - 12x + 11 - 2x^2 - 1 \\ 0 = 2x^2 - 12x + 10 \] Dividing by 2 simplifies this to: \[ 0 = x^2 - 6x + 5 \] Factoring yields: \[ 0 = (x - 5)(x - 1) \] Thus, the solutions are: \[ x = 5 \quad \text{and} \quad x = 1 \] So the correct answer is: **c. \( \quad x=5,1 \)**