Problem 64 - Elementary and Intermediate Algebra

Question

Let \(f(x)=3 x-2\) and \(g(x)=x^{2}+x .\) Find each of the following. $$ (f \circ g)(x) $$

Step-by-Step Solution

Answer

The expression \((f \circ g)(x)\) is \(3x^2 + 3x - 2\).

Step 1: Understand the Composition

The function notation \((f \circ g)(x)\) means we need to find the value of the function \(f\) when the output of the function \(g\) is input into \(f\). This involves substituting \(g(x)\) into \(f(x)\).

Step 2: Substitute g(x) into f(x)

We need to replace every \(x\) in \(f(x) = 3x - 2\) with \(g(x)\), which is \(x^2 + x\). Therefore, substitute \(g(x)\) into \(f\), resulting in \(f(g(x)) = 3(x^2 + x) - 2\).

Step 3: Simplify the Expression

Distribute the 3 through the expression \(3(x^2 + x) - 2\). This results in \(3x^2 + 3x - 2\).

Step 4: Write Final Expression

The expression \((f \circ g)(x)\) is now simplified to \(3x^2 + 3x - 2\).

Key Concepts

Understanding Polynomial Functions

Polynomials are fundamental expressions in algebra. They are composed of variables, coefficients, and exponents, using basic operations such as addition, subtraction, and multiplication. A polynomial function is typically written as:

• Constant terms (e.g., 5)
• Linear terms (e.g., \(3x\))
• Quadratic terms (e.g., \(x^2\))
• Cubic terms (e.g., \(x^3\)), and so on.

Each term in a polynomial is a product of a constant and a variable raised to a whole number exponent. The highest exponent in the polynomial determines its degree, an important characteristic that impacts the graph's shape and behavior. So when solving problems involving polynomials, recognizing the degree and the types of terms helps in comprehending and manipulating the expressions effectively.
In the exercise, the function \(g(x) = x^2 + x\) is a polynomial function with a quadratic term \(x^2\). It's crucial to understand that you can perform operations like adding, subtracting, or substituting these polynomials just like numerical expressions.

Decoding Function Notation

Function notation is a compact way to express the input-output relationship in mathematics. When you see a symbol like \(f(x)\), it indicates a function named \(f\) with \(x\) as the variable input. Such notations allow mathematicians to write complex equations simply, focusing on relationships instead of mere calculations.
An important aspect is understanding composition, denoted by \((f \circ g)(x)\). This means you apply function \(g\) first and then take its result as an input to function \(f\). Essentially, you're layering functions, much like a nesting doll. This composition creates a new function that conveys the entire process in one expression.
For example, in the given exercise, \(f(g(x))\) means substituting \(g(x)\) into \(f(x)\). It's a seamless process that consolidates two functions into a single operational statement.

Exploring Algebraic Expressions and Their Manipulations

Algebraic expressions form the backbone of algebra, a branch of mathematics that uses symbols to represent numbers and operations. A simple algebraic expression might look like \(3x^2 + 3x - 2\), made up of terms consisting of constants and variables.
One major skill in manipulating algebraic expressions is substitution. This involves replacing a variable with another expression or value. In function composition, this skill is vital as seen when substituting \(g(x)\) for \(x\) in \(f(x)\). It transforms the original function into a new expression with ease.
When simplifying expressions, understanding distributive properties helps distribute terms correctly across the parentheses, efficiently breaking down complex expressions into simpler parts.
For instance, in the exercise, distributing the \(3\) across \(x^2 + x\) results in \(3x^2 + 3x\), which is then adjusted by subtracting \(2\), leading to the final combined expression: \(3x^2 + 3x - 2\). Knowing these manipulations solidifies the foundation for solving a vast array of algebraic and functional problems.