In Exercises 51–66, find a. (fog) (x) b. (go f) (x) f(x) = x²+2, g(x) = x² – 2

Verified step by step guidance

1. Step 1: Understand the notation (f ∘ g)(x) and (g ∘ f)(x). This means composing the functions f and g. Specifically, (f ∘ g)(x) means f(g(x)) and (g ∘ f)(x) means g(f(x)).
2. Step 2: To find (f ∘ g)(x), substitute g(x) into f(x). Since f(x) = x² + 2 and g(x) = x² - 2, replace every x in f(x) with g(x).
3. Step 3: Calculate f(g(x)) by substituting g(x) = x² - 2 into f(x) = x² + 2, resulting in f(g(x)) = (x² - 2)² + 2.
4. Step 4: To find (g ∘ f)(x), substitute f(x) into g(x). Replace every x in g(x) with f(x).
5. Step 5: Calculate g(f(x)) by substituting f(x) = x² + 2 into g(x) = x² - 2, resulting in g(f(x)) = (x² + 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)). Conversely, (go f)(x) means applying f first and then g, written as g(f(x)). Understanding this concept is crucial for solving the given problem.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax² + bx + c. In this question, f(x) = x² + 2 and g(x) = x² - 2 are both quadratic functions. Recognizing their structure helps in performing operations like composition, as the resulting functions will also be quadratic.

Evaluating Functions

Evaluating functions involves substituting a specific value into the function to find the output. For example, to evaluate f(3) for f(x) = x² + 2, you would calculate 3² + 2 = 11. This skill is essential when working with composed functions, as you will need to evaluate the inner function before applying the outer function.