Find a. (fog) (x) b. (go f) (x) f(x)=4x-3, g(x) = 5x² - 2

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Step 1: Understand the problem. You are tasked with finding two compositions of functions: (f ∘ g)(x) and (g ∘ f)(x). The notation (f ∘ g)(x) means 'f of g of x,' or f(g(x)), and (g ∘ f)(x) means 'g of f of x,' or g(f(x)).

Step 2: Start with (f ∘ g)(x). Substitute g(x) into f(x). Since f(x) = 4x - 3 and g(x) = 5x² - 2, replace the 'x' in f(x) with g(x). This gives f(g(x)) = 4(5x² - 2) - 3.

Step 3: Simplify the expression for (f ∘ g)(x). Distribute the 4 across the terms inside the parentheses: 4(5x² - 2) becomes 20x² - 8. Then subtract 3 to get 20x² - 8 - 3. Combine like terms to simplify further.

Step 4: Now, find (g ∘ f)(x). Substitute f(x) into g(x). Since g(x) = 5x² - 2 and f(x) = 4x - 3, replace the 'x' in g(x) with f(x). This gives g(f(x)) = 5(4x - 3)² - 2.

Step 5: Simplify the expression for (g ∘ f)(x). Expand (4x - 3)² using the formula (a - b)² = a² - 2ab + b². Then multiply the result by 5 and subtract 2. Combine like terms to simplify further.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Evaluating Functions

Evaluating functions requires substituting a specific value into the function's formula to find the output. For example, to evaluate f(x) = 4x - 3 at x = 2, you would calculate f(2) = 4(2) - 3 = 5. This skill is essential for determining the values of (fog)(x) and (go f)(x) in the exercise.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form g(x) = ax² + bx + c. In this problem, g(x) = 5x² - 2 is a quadratic function, and understanding its properties, such as its graph's parabolic shape and vertex, is important for analyzing the composition with f(x).

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