Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Find (a) \(f \circ g\) and (b) \(g \circ f\). \(f(x)=\frac{1}{2} x+1, \quad g(x)=2 x+3\)

Step-by-Step Solution

Answer

The composite function \(f(g(x)) = x + \frac{5}{2}\) and the composite function \(g(f(x)) = x + 5\).

Step 1: Calculation of \(f \circ g\)

Substitute \(g(x) = 2x+3\) into \(f(x)\) to get \(f(g(x))\). Hence, \(f(g(x)) = f(2x+3) = \frac{1}{2} (2x+3) + 1 = x + \frac{3}{2} + 1 = x + \frac{5}{2}\)

Step 2: Calculation of \(g \circ f\)

Substitute \(f(x) = \frac{1}{2}x + 1\) into \(g(x)\) to get \(g(f(x))\). Hence, \(g(f(x)) = g(\frac{1}{2}x + 1) = 2(\frac{1}{2}x + 1) + 3 = x + 2 + 3 = x + 5\)

Key Concepts

Function Operations

Understanding function operations is foundational in algebra and precalculus. Function operations include addition, subtraction, multiplication, division, and composition. Just as with numbers, you can perform these operations on functions to create new functions. For instance, given two functions, f(x) and g(x), we can add them to create a new function (f + g)(x) = f(x) + g(x). Similarly, we could find their difference (f - g)(x), product (f * g)(x), or their quotient (f / g)(x), provided g(x) is not zero at points of interest.

In the textbook problem provided, we focus on another fundamental operation called composition, denoted by \( f \circ g \), which is the focus of our exploration. The process involves replacing the input of one function with another function, thus creating a sequence of operations. This is a critical concept to grasp as it forms the basis of more complex functions encountered in higher mathematics.

Algebraic Functions

An algebraic function is a type of function that can be written using algebraic expressions. These expressions are built from constants, variables, and the arithmetic operations (addition, subtraction, multiplication, division, and taking exponents). Algebraic functions can be simple linear functions like f(x) = 2x + 1, or more complex ones involving polynomials, radicals, or rational expressions. The functions in our exercise, \( f(x) = \frac{1}{2} x + 1 \) and \( g(x) = 2x + 3 \) are examples of linear functions, which are the most basic type of algebraic functions.

Linear functions have constant rates of change and graph as straight lines. Recognizing different types of algebraic functions and understanding their behaviors is crucial for solving equations and inequalities, as well as analyzing function operations like composition.

Function Composition

Function composition is a way to combine two functions into a single function. The notation \( f \circ g \) reads 'f composed with g' and represents a situation where the output of g(x) becomes the input of f(x). To find the composition of the functions, you replace every occurrence of x in f(x) with the function g(x).

The textbook problem illustrates the concept with two linear functions. Step 1 shows the composition \( f \circ g \) by substituting g(x) into f(x), which yields the function \( x + \frac{5}{2} \) after simplifying. Similarly, in step 2 for \( g \circ f \) we substitute f(x) into g(x), resulting in a new function \( x + 5 \) after simplification.

Remember, the order in which functions are composed matters. That is, \( f \circ g \) is generally not the same as \( g \circ f \) as clearly demonstrated by the different results in each of the steps from the provided exercise. Comprehending this concept and being able to compute compositions is crucial for understanding more advanced concepts like inverse functions and transformations of functions.