Find a. (fog) (x) b. (go f) (x) c. (fog) (2) d. (go f) (2).
f(x) = 1/x, g(x)= 1/x

Verified step by step guidance

1. Step 1: Understand the composition of functions. The notation (f ∘ g)(x) means f(g(x)), which involves substituting g(x) into f(x). Similarly, (g ∘ f)(x) means g(f(x)), which involves substituting f(x) into g(x).
2. Step 2: For part (a), calculate (f ∘ g)(x). Substitute g(x) = 1/x into f(x) = 1/x. This gives f(g(x)) = f(1/x). Replace x in f(x) with 1/x to get the expression for (f ∘ g)(x).
3. Step 3: For part (b), calculate (g ∘ f)(x). Substitute f(x) = 1/x into g(x) = 1/x. This gives g(f(x)) = g(1/x). Replace x in g(x) with 1/x to get the expression for (g ∘ f)(x).
4. Step 4: For part (c), evaluate (f ∘ g)(2). Use the expression for (f ∘ g)(x) derived in part (a) and substitute x = 2. Simplify the resulting expression.
5. Step 5: For part (d), evaluate (g ∘ f)(2). Use the expression for (g ∘ f)(x) derived in part (b) and substitute x = 2. Simplify the resulting expression.

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)). This concept is essential for solving the given exercises, as it requires understanding how to manipulate and evaluate the functions in sequence.

Evaluating Functions

Evaluating functions means substituting a specific value into the function to find the output. For example, if f(x) = 1/x, evaluating f(2) involves substituting 2 for x, resulting in f(2) = 1/2. This skill is crucial for calculating the values of (fog)(2) and (go f)(2) in the exercises.

Reciprocal Functions

Reciprocal functions are functions of the form f(x) = 1/x, which are defined for all x except zero. They exhibit unique properties, such as having vertical asymptotes at x = 0 and being symmetric about the line y = x. Understanding the behavior of reciprocal functions is important for accurately performing the compositions and evaluations in the problem.