In Exercises 51–66, 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 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: To find $(f\circ g)(x)$, substitute $g(x)$ into $f(x)$. So, replace $x$ in $f(x)=2x-3$ with $g(x)=\frac{x+3}{2}$.
3. Step 3: Simplify the expression $f(g(x))=2\left(\frac{x+3}{2}\right)-3$.
4. Step 4: To find $(g\circ f)(x)$, substitute $f(x)$ into $g(x)$. So, replace $x$ in $g(x)=\frac{x+3}{2}$ with $f(x)=2x-3$.
5. Step 5: Simplify the expression $g(f(x))=\frac{(2x-3)+3}{2}$.

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 value into a function to find its output. For example, if f(x) = 2x - 3, evaluating f(2) involves replacing x with 2, resulting in f(2) = 2(2) - 3 = 1. This skill is essential for calculating the values of composite functions at given points.

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 for simplifying composite functions and performing evaluations accurately.