Find a. (fog) (2) b. (go f) (2) 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)(2) and (g ∘ f)(2). This means you will substitute one function into the other and evaluate the result at x = 2.

Step 2: Recall the definition of function composition. For (f ∘ g)(x), this means f(g(x)). For (g ∘ f)(x), this means g(f(x)).

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

Step 4: Evaluate (f ∘ g)(2). Substitute x = 2 into the simplified expression from Step 3. Perform the arithmetic to find the result.

Step 5: Now, find (g ∘ f)(2). Substitute f(x) into g(x). Replace x in g(x) = 5x² - 2 with f(x) = 4x - 3. This gives g(f(x)) = 5(4x - 3)² - 2. Simplify this expression, then substitute x = 2 and evaluate.

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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)). 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, to evaluate f(2) for f(x) = 4x - 3, you would replace x with 2, resulting in f(2) = 4(2) - 3 = 5. This skill is essential for calculating the values of (fog)(2) and (go f)(2) 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 shape (a parabola) and how it interacts with linear functions, is important for correctly performing the compositions and evaluations.

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