This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Understand the given functions. We are given two functions: f(x) = x^2 + 1 g(x) = x + 1 Step 2: Solve part a). a) Find (f g)(1). This means f(g(1)). First, evaluate g(1): g(1) = 1 + 1 = 2 Next, evaluate f(2): f(2) = (2)^2 + 1 = 4 + 1 = 5 So, (f g)(1) = 5. Step 3: Solve part b). b) Find (g f)(1). This means g(f(1)). First, evaluate f(1): f(1) = (1)^2 + 1 = 1 + 1 = 2 Next, evaluate g(2): g(2) = 2 + 1 = 3 So, (g f)(1) = 3. Step 4: Solve part c). c) Find (f g)(x). This means f(g(x)). Substitute g(x) into f(x): f(g(x)) = f(x+1) Now, substitute (x+1) into the expression for f(x): f(x+1) = (x+1)^2 + 1 Expand the expression: f(x+1) = (x^2 + 2x + 1) + 1 f(x+1) = x^2 + 2x + 2 So, (f g)(x) = x^2 + 2x + 2. Step 5: Solve part d). d) Find (g f)(x). This means g(f(x)). Substitute f(x) into g(x): g(f(x)) = g(x^2+1) Now, substitute (x^2+1) into the expression for g(x): g(x^2+1) = (x^2+1) + 1 g(x^2+1) = x^2 + 2 So, (g f)(x) = x^2 + 2. Step 6: Solve part e). e) Find (f g)(a^2). This means f(g(a^2)). First, evaluate g(a^2): g(a^2) = a^2 + 1 Next, evaluate f(a^2+1): f(a^2+1) = (a^2+1)^2 + 1 Expand the expression: f(a^2+1) = (a^4 + 2a^2 + 1) + 1 f(a^2+1) = a^4 + 2a^2 + 2 So, (f g)(a^2) = a^4 + 2a^2 + 2. Step 7: Solve part f). f) Find (g f)(sqrt(a)). This means g(f(sqrt(a))). First, evaluate f(sqrt(a)): f(sqrt(a)) = (sqrt(a))^2 + 1 = a + 1 Next, evaluate g(a+1): g(a+1) = (a+1) + 1 = a + 2 So, (g f)(sqrt(a)) = a + 2.