Problem 27 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The composite functions are \(f \circ g = 9x^2 + 6x + 1\), \(g \circ f = 3x^2 + 1\), and \(f \circ f = x^4\).

Step 1: Find \(f \circ g\)

Substitute \(g(x)\) into \(f(x)\). This means wherever \(x\) appears in \(f(x)\), replace it with \(g(x)\).\n\So, \(f \circ g = f(g(x)) = (3x +1)^2 = 9x^2 + 6x + 1\).

Step 2: Find \(g \circ f\)

Substitute \(f(x)\) into \(g(x)\). This means wherever \(x\) appears in \(g(x)\), replace it with \(f(x)\).\n\So, \(g \circ f = g(f(x)) = 3(x^2) + 1 = 3x^2 + 1\).

Step 3: Find \(f \circ f\)

Substitute \(f(x)\) into itself. This means wherever \(x\) appears in \(f(x)\), replace it with \(f(x)\).\n\So, \(f \circ f = f(f(x)) = (x^2)^2 = x^4\).

Key Concepts

Algebraic Functions

Algebraic functions are mathematical expressions that involve operations of addition, subtraction, multiplication, division, and taking roots with variables. They can range from simple linear functions to more complex polynomial expressions. When we talk about algebraic functions like \( f(x) = x^2 \) and \( g(x) = 3x + 1 \), we are dealing with functions that operate on variables in such a way that they transform inputs into outputs according to the rules defined by these operations.
Understanding the composition of algebraic functions is key to solving problems like the one given in the original exercise. Function composition involves taking the output of one function and using it as the input for another. This idea is not only fundamental in algebra but also critical in understanding more complex systems of equations.

• Composition Operation: For two functions \( f \) and \( g \), the composite function \( f \, \circ \, g \) is defined as \( f(g(x)) \).
• Application: When composing \( f \) and \( g \), replace all instances of the variable in \( f(x) \) with \( g(x) \).

Exploring this concept with algebraic functions provides a structured way to analyze how functions interact and transform values. This is especially useful for learning more advanced topics like calculus.

Polynomial Functions

Polynomial functions are a specific type of algebraic function characterized by expressions involving powers of a variable. Each term in a polynomial function has a coefficient and an exponent, with the general form being \( a_n x^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0 \), where each \( a_i \) is a constant coefficient.
In the original problem,\( f(x) = x^2 \) and the product \( (3x+1)^2 \) are examples of polynomial functions. Let's break them down further:

• Degree: The degree of a polynomial function is the highest power of its variable. For instance, \( f(x) = x^2 \) is a second-degree polynomial, while \( (3x + 1)^2 = 9x^2 + 6x + 1 \) is also quadratic or second-degree in nature.
• Coefficients: In the polynomial \( f(x) = x^2 \), the coefficient of the \( x^2 \) term is 1. For \( g(x) = 3x + 1 \) after expansion, it becomes \( 9x^2 + 6x + 1 \), where the coefficients are 9, 6, and 1 respectively.

Polynomial functions are not only fundamental in algebra but also pervasive in various applications across science and engineering. They are crucial for modeling relationships and can be easily analyzed for their roots and other properties.

Mathematical Notation

Mathematical notation is a system of symbols and signs used to represent numbers, expressions, and operations. It helps in creating a universally understood language for mathematicians and students.
In the context of the problem, we use notation to denote:

• Functions: Represented usually as \( f(x) \), indicates a function \( f \) operating on variable \( x \).
• Composition: The symbol \( \circ \) indicates function composition, where \( f \circ g = f(g(x)) \).
• Exponentiation: The symbol \( ^n \) is used to denote a variable \( x \) raised to the power of \( n \).

By using these symbols, mathematical notation enables clear communication of complex ideas with precision and brevity. Understanding these conventions is pivotal to being proficient in algebra and can significantly enhance comprehension in problem-solving. Embracing proper notation also ensures that solutions can be universally shared and verified across different contexts and problems.