Problem 6 - College Algebra

Question

Find the domain of each function. $$ f(x)-x^{2}+x-12 $$

Step-by-Step Solution

Answer

The domain of the function \( f(x) = x^{2} + x - 12 \) is all real numbers.

Step 1: Identify the function type

The function \( f(x) = x^{2} + x - 12 \) is a polynomial function of second degree.

Step 2: Analyze the function

In a polynomial function, the y values (function outputs) are determined by real number x values (function inputs). Meaning, there's no restriction on x, the function is defined for every real x.

Step 3: Determine the domain

Since there is no restriction, it can be said that the function is defined for all real x. Thus, the domain of this function is all real numbers.

Key Concepts

Polynomial Functions

When studying polynomial functions, it's important to grasp their basic characteristics and structures. A polynomial function is composed of terms that are simply non-negative integer powers of the variable (usually 'x') multiplied by coefficients. These can take a wide variety of shapes, from a simple linear function like f(x) = x, to a quadratic function such as f(x) = x^2, up to higher degrees.

One notable feature of polynomial functions is their smooth, continuous nature. They do not have breaks, jumps, or sharp corners, which makes analyzing them quite straightforward in most cases. These functions are also well-behaved at infinity: as 'x' gets very large, the term with the highest power of 'x' usually dominates the behavior of the function.

A second-degree polynomial, often called a quadratic, has the general form f(x) = ax^2 + bx + c. In the example f(x) = x^2 + x - 12, 'a' is 1, 'b' is 1, and 'c' is -12. This tells us a lot about the function, including the potential for a parabolic graph and a well-defined vertex.

Real Numbers

The concept of real numbers is fundamental to understanding the domain of functions. Real numbers include all the numbers on the number line, encompassing both rational numbers (such as 6, 0.5, and -2/3) and irrational numbers (like \( \sqrt{2} \) and \( \pi \)). Essentially, if you can represent it as a point on the number line, it's a real number.

When analyzing the domain of a function, we are looking at the set of all possible input values. For polynomial functions, the domain is usually all real numbers because you can input any real number into the function and receive a real number output. There are no values for 'x' in a polynomial function that would result in a non-real or undefined outcome, which is a critical point for students to understand when determining domains.

Function Analysis

The step-by-step process of function analysis allows us to inspect characteristics such as domain, range, continuity, and behavior at extremes. Starting with the domain, as was done in the given solution, is one of the first steps in function analysis.

For instance, taking the function f(x) = x^2 + x - 12, we determine its domain by looking for any x-values that could cause issues, such as division by zero or square roots of negative numbers. Since there are none, the domain of this quadratic function is all real numbers.

Advanced function analysis might also include finding intercepts, turning points, end behavior, and asymptotes in relevant functions. This level of detail gives a complete picture of a function's behavior across its domain, but for polynomial functions like the one examined here, the simplicity of their structure allows us to easily conclude that their domain is all real numbers.