Composition of Functions Worksheet Name \( \qquad \) Ryl. I. Let \( f(x)=2 x-1, g(x)=3 x \), and \( h(x)=x^{2}+1 \). Comput 1. \( f(g(x)) \) \( 2(3 x)-1 \) \( 6 x-1 \) 2. \( f(h(x)) \) \( 2\left(x^{2}+1\right)-1 \) 4. \( h(f(9)) \) 6. \( 1\binom{4}{4} \) 5. \( g(f(0)) \) \[ 18-1=17 \] \[ h(17)=17^{2}+1 \] 7. \( f(g(h(2))) \) 8. \( h(g(f(5))) \)

Let's solve the composition of functions step by step based on the given functions \( f(x) = 2x - 1 \), \( g(x) = 3x \), and \( h(x) = x^2 + 1 \). ### 1. Compute \( f(g(x)) \) We already have: \[ f(g(x)) = f(3x) = 2(3x) - 1 = 6x - 1 \] ### 2. Compute \( f(h(x)) \) Now, let's compute \( f(h(x)) \): \[ f(h(x)) = f(x^2 + 1) = 2(x^2 + 1) - 1 = 2x^2 + 2 - 1 = 2x^2 + 1 \] ### 3. Compute \( h(f(9)) \) First, we need to find \( f(9) \): \[ f(9) = 2(9) - 1 = 18 - 1 = 17 \] Now, we compute \( h(17) \): \[ h(17) = 17^2 + 1 = 289 + 1 = 290 \] ### 4. Compute \( g(f(0)) \) First, we find \( f(0) \): \[ f(0) = 2(0) - 1 = -1 \] Now, we compute \( g(-1) \): \[ g(-1) = 3(-1) = -3 \] ### 5. Compute \( f(g(h(2))) \) First, we need to find \( h(2) \): \[ h(2) = 2^2 + 1 = 4 + 1 = 5 \] Next, we compute \( g(5) \): \[ g(5) = 3(5) = 15 \] Finally, we compute \( f(15) \): \[ f(15) = 2(15) - 1 = 30 - 1 = 29 \] ### 6. Compute \( h(g(f(5))) \) First, we find \( f(5) \): \[ f(5) = 2(5) - 1 = 10 - 1 = 9 \] Next, we compute \( g(9) \): \[ g(9) = 3(9) = 27 \] Finally, we compute \( h(27) \): \[ h(27) = 27^2 + 1 = 729 + 1 = 730 \] ### Summary of Results 1. \( f(g(x)) = 6x - 1 \) 2. \( f(h(x)) = 2x^2 + 1 \) 3. \( h(f(9)) = 290 \) 4. \( g(f(0)) = -3 \) 5. \( f(g(h(2))) = 29 \) 6. \( h(g(f(5))) = 730 \) These are the results for the compositions of the functions as requested.