Find (a) \( f(g(x)) \), (b) \( g(f(x)) \), and (c) \( f(f(x)) \). \( f(x)=2 x+1, g(x)=x^{2}+6 x-10 \) a. \( f(g(x))=\square \) b. \( g(f(x))=\square \) c. \( f(f(x))=\square \)

To find the compositions of the functions \( f(x) \) and \( g(x) \), we can follow these steps: (a) To find \( f(g(x)) \): - We first calculate \( g(x) \) which is \( g(x) = x^2 + 6x - 10 \). - Now, substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(x^2 + 6x - 10) = 2(x^2 + 6x - 10) + 1 = 2x^2 + 12x - 20 + 1 = 2x^2 + 12x - 19. \] So, \( f(g(x)) = 2x^2 + 12x - 19 \). (b) To find \( g(f(x)) \): - We substitute \( f(x) \) into \( g(x) \): \[ g(f(x)) = g(2x + 1) = (2x + 1)^2 + 6(2x + 1) - 10. \] Calculating this step by step: 1. \( (2x + 1)^2 = 4x^2 + 4x + 1 \) 2. \( 6(2x + 1) = 12x + 6 \) 3. Combine these: \( 4x^2 + 4x + 1 + 12x + 6 - 10 = 4x^2 + 16x - 3 \). Thus, \( g(f(x)) = 4x^2 + 16x - 3 \). (c) To find \( f(f(x)) \): - Substitute \( f(x) \) into itself: \[ f(f(x)) = f(2x + 1) = 2(2x + 1) + 1 = 4x + 2 + 1 = 4x + 3. \] So, \( f(f(x)) = 4x + 3 \). In summary: a. \( f(g(x)) = 2x^2 + 12x - 19 \) b. \( g(f(x)) = 4x^2 + 16x - 3 \) c. \( f(f(x)) = 4x + 3 \)