Problem 28 - Elementary and Intermediate Algebra

Question

Let \(f(x)=3 x\) and \(g(x)=4 x .\) Find each function and give its domain. $$ g \cdot f $$

Step-by-Step Solution

Answer

The function \((g \cdot f)(x) = 12x^2\) with a domain of all real numbers \(\mathbb{R}\).

Step 1: Identify the Functions

The functions provided are \(f(x) = 3x\) and \(g(x) = 4x\). These are both linear functions with an implicit domain of all real numbers (\(\mathbb{R}\)) because there are no restrictions (like division by zero or square roots of negative numbers) on \(x\) for these functions.

Step 2: Perform Function Multiplication

To find \((g \cdot f)(x)\), you need to multiply the two functions: \[ (g \cdot f)(x) = g(x) \cdot f(x) = (4x) \cdot (3x) = 12x^2 \] Thus, the product of these functions is \(12x^2\).

Step 3: Determine the Domain of the Product Function

The domain of \(12x^2\) is determined by the restrictions on \(x\). Since \(12x^2\) is a polynomial, and polynomials have domains of all real numbers, the domain of \((g \cdot f)(x) = 12x^2\) is also all real numbers, \(\mathbb{R}\).

Key Concepts

Linear Functions

Functions like \( f(x) = 3x \) and \( g(x) = 4x \) are called linear functions. They are simple, straight lines when graphed, where each function can be written in the form \( f(x) = ax + b \). In the case of our functions, both \( a \) is 3 for \( f(x) \) and 4 for \( g(x) \), while \( b \), which represents the y-intercept, is zero. This means both lines pass through the origin (0,0) on a graph.

Linear functions are crucial because they offer a direct and proportional relationship between \( x \) and \( f(x) \). Each increase in \( x \) results in a predictable increase in \( f(x) \). It's like moving up or down a hill with a constant slope.

• The slope \( a \) tells us how steep the line is.
• If the slope \( a \) is positive, the line rises from left to right.
• When graphed, every linear function creates a straight line.

Understanding linear functions allows you to predict and calculate outcomes using simple arithmetic.

Polynomial Domain

The domain of a polynomial function, such as \( 12x^2 \), is determined by which inputs for \( x \) provide valid outputs for the function.
Polynomials are algebraic expressions that include terms with powers of \( x \). They can be as simple as a linear expression or contain higher degree terms (like squared or cubic terms).

Here are the key features you need to know about polynomial domains:

• Polynomials are typically defined for all real numbers, \( \mathbb{R} \).
• There's no risk of division by zero or negative square root issues in polynomial expressions.
• This includes the function \( 12x^2 \) as it is simply \( x \) squared times a constant.

With no restrictions, you can plug any real number into a polynomial and receive a real number output, making them versatile for mathematical modeling and predictions.

Real Numbers

The realm of real numbers \( \mathbb{R} \) covers all numbers that can be located on a standard number line. This includes both rational numbers (like fractions and integers) and irrational numbers (like \( \pi \) and \( \sqrt{2} \)).

Real numbers play an essential role in mathematics since they provide a comprehensive framework for understanding quantities, distances, and measurements.

• Rational numbers have clear rates and repeating patterns when expressed as decimals.
• Irrational numbers have non-repeating, non-terminating decimal expansions.
• Every point on the number line corresponds to a real number.

Function domains often include all real numbers. In our exercise, since there's no restriction given by fractional or square root operation, linear and polynomial functions include \( \mathbb{R} \) in their domain. Understanding real numbers prepares you to handle diverse mathematical scenarios.