Problem 65 - Elementary Algebra

Question

Calculate \((f \cdot g)(x)\) and determine the restrictions to the domain. $$ f(x)=1 x \text { and } g(x)=1 x-1 $$

Step-by-Step Solution

Answer

\((f \cdot g)(x) = x^2 - x\); no domain restrictions, domain is all real numbers.

Step 1: Understand the Problem

We need to find the product of the functions \( f(x) \) and \( g(x) \), which is \( (f \cdot g)(x) = f(x) \times g(x) \). Then, we need to determine any restrictions, if applicable, on the domain of this product function.

Step 2: Write Down the Functions

The functions provided are \( f(x) = 1x = x \) and \( g(x) = 1x - 1 = x - 1 \).

Step 3: Multiply the Functions

We now compute the product \( (f \cdot g)(x) \) by multiplying \( f(x) \) and \( g(x) \). This gives:\[(f \cdot g)(x) = x \times (x - 1).\]

Step 4: Expand the Product

Expand the expression from Step 3:\[(f \cdot g)(x) = x \times (x - 1) = x^2 - x.\]

Step 5: Determine Domain Restrictions

Consider potential restrictions. Both \( f(x) \) and \( g(x) \) are polynomial functions, which are defined for all real numbers. Therefore, \( x \) can be any real number.

Step 6: Finalize the Solution

The final expression for the product is \( (f \cdot g)(x) = x^2 - x \). Since polynomials have no undefined points or restrictions, the domain is all real numbers.

Key Concepts

Function Multiplication

Function multiplication involves taking two functions, say \( f(x) \) and \( g(x) \), and creating a new function by multiplying them together. This is denoted as \((f \cdot g)(x)\). In simple terms, this means

• replacing the input \( x \) into each function,
• computing each function, and then
• multiplying the outputs together.

For the given functions \( f(x) = x \) and \( g(x) = x - 1 \), their product function becomes \[(f \cdot g)(x) = x \cdot (x - 1) = x^2 - x.\] This means at each value of \( x \), you substitute into \( x \) and \( x - 1 \), calculate each, and then multiply the results to obtain a new value. This process of multiplying functions allows us to explore new relationships between variables, providing a clearer picture of how they interact with each other.

Domain Restrictions

In most cases, when we multiply functions, it's important to consider any domain restrictions. These are values for which the function is not defined. However, polynomial functions, like \( x^2 - x \), do not have any restrictions.

• Unlike fractions or square roots, polynomials are continuously defined over all real numbers.
• We simply need to check if there’s a possibility of undefined behavior.

Since both \( f(x) \) and \( g(x) \) are linear, there are no divisions by zero or negative square roots to consider. Thus, the domain here is all real numbers \( \mathbb{R} \), and there are no restrictions for the product \((f \cdot g)(x) = x^2 - x\). This universal domain of polynomial functions makes them quite straightforward and predictable, posing no intrinsic boundaries.

Polynomial Functions

Polynomial functions are expressively versatile in mathematics because they handle functions defined for any real number. A polynomial function has terms of varying powers of the variable, usually written as \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \). The degree of a polynomial,

• is determined by the highest power of \( x \).
• For \( x^2 - x \), the degree is 2, making it a quadratic polynomial.

This particular polynomial, \( x^2 - x \), gives us a parabolic shape on a graph. Importantly, polynomials like this one are

• continuous,
• smooth to graph, and
• possess no breaks or holes.

Such characteristics make them invaluable in modeling relationships in algebra and calculus. Understanding polynomial functions helps students gain insights into more complex algebraic expressions, ensuring they grasp foundational concepts in mathematics seamlessly.