Problem 38 - Precalculus with Limits

Question

In Exercises 37-40, find (a) \(f \circ g\), (b) \(g \circ f\), and (c) \(g \circ g\). \(f(x) = 3x + 5\), \(g(x) = 5 - x\)

Step-by-Step Solution

Answer

The results of the function compositions are: (a) \(f \circ g(x) = 20 - 3x\), (b) \(g \circ f(x) = -3x\), and (c) \(g \circ g(x) = x\).

Step 1: Compute \(f \circ g(x)\)

Function composition \(f \circ g(x)\) means that we substitute \(g(x)\) into \(f(x)\). Given \(f(x) = 3x + 5\) and \(g(x) = 5 - x\), our composition becomes \(f(g(x)) = 3(5 - x) + 5 = 15 - 3x + 5 = 20 - 3x\).

Step 2: Compute \(g \circ f(x)\)

Function composition \(g \circ f(x)\) means that we substitute \(f(x)\) into \(g(x)\). Given \(f(x) = 3x + 5\) and \(g(x) = 5 - x\), our composition becomes \(g(f(x)) = 5 - (3x + 5) = 5 - 3x - 5 = -3x\).

Step 3: Compute \(g \circ g(x)\)

Function composition \(g \circ g(x)\) means that we substitute \(g(x)\) for x in the expression of \(g(x)\). Our composition becomes \(g(g(x)) = 5 - (5 - x) = 5 - 5 + x = x\).

Key Concepts

Algebraic Functions

Algebraic functions are mathematical expressions that involve polynomials, which include variables raised to whole number powers. Commonly, algebraic functions can be seen in the form of linear functions, which are first-degree polynomials. For example, consider the function given in the exercise, such as * \( f(x) = 3x + 5 \) In this function:

• The coefficient "3" multiplies the variable "x"
• "+ 5" is a constant term added to the product

These components represent a straight line when graphed on the coordinate plane.
Understanding how to manipulate and combine such functions is foundational in mathematics, which leads us to their composition.

Composite Functions

Composite functions involve combining two or more functions to form a new function. This is done by substituting one function into another, which is particularly useful in solving more complex problems. When combining functions, the notation * \( f \circ g \) read as "f composed with g." It signifies substituting the function \( g(x) \) into \( f(x) \).

• For instance, in the exercise, \( f(g(x)) = 20 - 3x \) derives from substituting \( g(x) = 5 - x \) into \( f(x) = 3x + 5 \).
• The reverse composition, \( g(f(x)) \), results in \( -3x \) by placing \( f(x) = 3x + 5 \) into \( g(x) = 5 - x \).

Each operation uses substitution to find a new rule for handling potential input values. The concept is intuitive but requires practice to ensure the clean insertion of one expression into another without errors.

Mathematical Notation

Mathematical notation is crucial for effectively communicating complex ideas succinctly. In terms of function composition, it simplifies the process of understanding what operations are performed in a systematic manner. The use of symbols and notation provides clarity:

• The symbol \( \circ \) indicates function composition, making it clear which function is input into another.
• Round brackets like \( f(x) \) provide structure, demonstrating where variables and substitutions occur within the function's rules.

For example, the expression * \( f(g(x)) \) indicates that \( g(x) \) is placed inside \( f(x) \). Such notation helps condense verbal explanations into clear mathematical expressions, enabling easier manipulation and understanding. Proper comprehension of these notations is vital for engaging with advanced mathematical problems effectively.