Find a. (fog) (x) b. (go f) (x) c. (fog) (2) d. (go f) (2).
f(x) = 2x-3, g(x) = (x+3)/2

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding the compositions of two functions, f(x) = 2x - 3 and g(x) = (x + 3)/2, in four parts: (fog)(x), (gof)(x), (fog)(2), and (gof)(2). Composition of functions means substituting one function into another.
2. Step 2: To find (fog)(x), substitute g(x) into f(x). Replace every instance of 'x' in f(x) = 2x - 3 with g(x) = (x + 3)/2. This gives f(g(x)) = 2((x + 3)/2) - 3. Simplify the expression.
3. Step 3: To find (gof)(x), substitute f(x) into g(x). Replace every instance of 'x' in g(x) = (x + 3)/2 with f(x) = 2x - 3. This gives g(f(x)) = ((2x - 3) + 3)/2. Simplify the expression.
4. Step 4: To find (fog)(2), use the result from Step 2 and substitute x = 2 into the simplified expression for (fog)(x). Simplify the result.
5. Step 5: To find (gof)(2), use the result from Step 3 and substitute x = 2 into the simplified expression for (gof)(x). Simplify the result.

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 evaluating the output of one function as the input for another.

Evaluating Functions

Evaluating functions means substituting a specific value into a function to find its output. For example, if f(x) = 2x - 3, to evaluate f(2), you replace x with 2, resulting in f(2) = 2(2) - 3 = 1. This skill is crucial for calculating the values of (fog)(2) and (go f)(2) in the exercises.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying expressions using algebraic rules. This includes operations like addition, subtraction, multiplication, and division of functions. Mastery of these techniques is necessary to simplify the results of function compositions and to ensure accurate evaluations in the exercises.