In Exercises 51–66, find a. (fog) (2) b. (go f) (2) f(x)=4x-3, g(x) = 5x² - 2

Verified step by step guidance

1. Step 1: Understand the composition of functions. The notation \((f \circ g)(x)\) means \(f(g(x))\), and \((g \circ f)(x)\) means \(g(f(x))\).
2. Step 2: For part (a), find \((f \circ g)(2)\). First, calculate \(g(2)\) using the function \(g(x) = 5x^2 - 2\). Substitute \(x = 2\) into \(g(x)\).
3. Step 3: Once you have \(g(2)\), substitute this result into the function \(f(x) = 4x - 3\) to find \(f(g(2))\).
4. Step 4: For part (b), find \((g \circ f)(2)\). First, calculate \(f(2)\) using the function \(f(x) = 4x - 3\). Substitute \(x = 2\) into \(f(x)\).
5. Step 5: Once you have \(f(2)\), substitute this result into the function \(g(x) = 5x^2 - 2\) to find \(g(f(2))\).

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)). Understanding this concept is crucial for solving problems that require evaluating composite functions.

Evaluating Functions

Evaluating functions means substituting a specific input value into a function to find its output. For example, to evaluate f(2) for f(x) = 4x - 3, you replace x with 2, resulting in f(2) = 4(2) - 3 = 5. This skill is essential for calculating the values of composite functions in the given exercises.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form g(x) = ax² + bx + c. In this case, g(x) = 5x² - 2 is a quadratic function. Understanding the properties of quadratic functions, such as their shape (parabola) and how they behave under composition, is important for solving the given problem.