Find c. (fog) (2) d. (go f) (2). f(x) = √x, g(x) = x − 1

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding two composite function values: (f ∘ g)(2) and (g ∘ f)(2). Composite functions involve substituting one function into another. The given functions are f(x) = √x and g(x) = x - 1.
2. Step 2: Start with (f ∘ g)(2). This means you first apply g(x) to the input 2, and then use the result as the input for f(x). Mathematically, (f ∘ g)(2) = f(g(2)).
3. Step 3: Calculate g(2). Substitute x = 2 into g(x) = x - 1. This gives g(2) = 2 - 1.
4. Step 4: Use the result of g(2) as the input for f(x). Substitute g(2) into f(x) = √x. This gives f(g(2)) = √(g(2)).
5. Step 5: Repeat the process for (g ∘ f)(2). This means you first apply f(x) to the input 2, and then use the result as the input for g(x). Mathematically, (g ∘ f)(2) = g(f(2)). Start by calculating f(2) = √2, and then substitute this result into g(x) = x - 1 to find g(f(2)).

Key Concepts

Function Composition

Function composition involves combining two functions to create a new function. If f and g are two functions, the composition (fog)(x) means applying g first and then applying 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, if f(x) = √x, to evaluate f(4), you would calculate √4, which equals 2. This skill is essential for determining the values of composite functions after performing the necessary compositions.

Square Root Function

The square root function, denoted as f(x) = √x, returns the non-negative value whose square is x. This function is defined only for non-negative inputs, making it important to consider the domain when evaluating or composing functions that include square roots. Understanding its properties helps in accurately solving problems involving this function.