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 problem. You are tasked with finding two compositions of functions: (f ∘ g)(x) and (g ∘ f)(x). This means you need to substitute one function into the other. The given functions are f(x) = x² + 2 and g(x) = x² - 2.
2. Step 2: To find (f ∘ g)(x), substitute g(x) into f(x). This means replacing every instance of 'x' in f(x) with the expression for g(x). Start with f(x) = x² + 2, and replace 'x' with g(x) = x² - 2. The resulting expression will be f(g(x)) = ((x² - 2)²) + 2.
3. Step 3: Simplify the expression for f(g(x)). Expand (x² - 2)² using the formula (a - b)² = a² - 2ab + b². This gives (x² - 2)² = x⁴ - 4x² + 4. Substitute this back into f(g(x)) to get f(g(x)) = x⁴ - 4x² + 4 + 2.
4. Step 4: To find (g ∘ f)(x), substitute f(x) into g(x). This means replacing every instance of 'x' in g(x) with the expression for f(x). Start with g(x) = x² - 2, and replace 'x' with f(x) = x² + 2. The resulting expression will be g(f(x)) = ((x² + 2)²) - 2.
5. Step 5: Simplify the expression for g(f(x)). Expand (x² + 2)² using the formula (a + b)² = a² + 2ab + b². This gives (x² + 2)² = x⁴ + 4x² + 4. Substitute this back into g(f(x)) to get g(f(x)) = x⁴ + 4x² + 4 - 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 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 case, f(x) = x² + 2 and g(x) = x² - 2 are both quadratic functions. Recognizing their structure helps in evaluating compositions and understanding their graphical representations, such as parabolas.

Evaluating Functions

Evaluating functions involves substituting a specific value of x into the function to find the corresponding output. For example, to find (fog)(x), you first calculate g(x) and then substitute that result into f. Mastery of this process is essential for accurately computing the compositions required in the exercises.