In Exercises 51–66, find a. (fog) (x) b. (go f) (x) c. (fog) (2) d. (go f) (2). f(x) = x+4, g(x) = 2x + 1

Verified step by step guidance

1. Step 1: Understand the composition of functions. The notation (f \circ g)(x) means f(g(x)), which means you substitute g(x) into f(x). Similarly, (g \circ f)(x) means g(f(x)).
2. Step 2: Find (f \circ g)(x). Substitute g(x) = 2x + 1 into f(x) = x + 4. So, f(g(x)) = (2x + 1) + 4.
3. Step 3: Simplify the expression from Step 2. Combine like terms to get the expression for (f \circ g)(x).
4. Step 4: Find (g \circ f)(x). Substitute f(x) = x + 4 into g(x) = 2x + 1. So, g(f(x)) = 2(x + 4) + 1.
5. Step 5: Simplify the expression from Step 4. Distribute and combine like terms to get the expression for (g \circ f)(x).

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 evaluating expressions where one function's output becomes 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) = x + 4, then f(2) = 2 + 4 = 6. This process is crucial for solving problems that require finding the value of a function at a given point, especially in the context of function composition.

Algebraic Manipulation

Algebraic manipulation refers to the techniques used to simplify or rearrange algebraic expressions. This includes operations like addition, subtraction, multiplication, and factoring. Mastery of these skills is necessary for effectively working with composed functions and ensuring accurate calculations when substituting values.