Find a. (fog) (2) b. (go f) (2) f(x) = x²+2, g(x) = x² – 2

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding two composite function values: (f ∘ g)(2) and (g ∘ f)(2). Composite functions involve substituting one function into another. The given functions are f(x) = x² + 2 and g(x) = x² - 2.
2. Step 2: Start with part (a), (f ∘ g)(2). This means you first evaluate g(2) and then substitute the result into f(x). To find g(2), substitute x = 2 into g(x): g(2) = (2)² - 2.
3. Step 3: After calculating g(2), substitute the result into f(x). Replace x in f(x) = x² + 2 with the value of g(2). This gives f(g(2)) = (g(2))² + 2.
4. Step 4: Move to part (b), (g ∘ f)(2). This means you first evaluate f(2) and then substitute the result into g(x). To find f(2), substitute x = 2 into f(x): f(2) = (2)² + 2.
5. Step 5: After calculating f(2), substitute the result into g(x). Replace x in g(x) = x² - 2 with the value of f(2). This gives g(f(2)) = (f(2))² - 2.

Key Concepts

Function Composition

Function composition involves combining two functions to create a new function. If f(x) and g(x) are two functions, the composition (fog)(x) means applying g first and then applying f to the result, expressed as f(g(x)). Understanding this concept is crucial for solving the given problem, as it requires evaluating the functions in a specific order.

Evaluating Functions

Evaluating functions means substituting a specific input value into the function's formula to find the output. For example, if f(x) = x² + 2, to evaluate f(2), you would calculate 2² + 2, resulting in 6. This skill is essential for finding the values of (fog)(2) and (go f)(2) in the exercise.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax² + bx + c. In this problem, both f(x) and g(x) are quadratic functions. Understanding their properties, such as their graphs being parabolas and their behavior under composition, is important for accurately calculating the compositions and their outputs.