Problem 44 - Algebra and Trigonometry

Question

Find the domain of the function. $$ f(x)=x^{2}+1 $$

Step-by-Step Solution

Answer

Domain: All real numbers \((-\infty, \infty)\).

Step 1: Understand the Function's Components

The function given is \( f(x) = x^2 + 1 \). This is a polynomial function, specifically a quadratic function, since the highest power of \( x \) is 2.

Step 2: Identify Domain Constraints

Polynomial functions, like \( f(x) = x^2 + 1 \), do not have any inherent restrictions on their domain. This means that no matter what real number value you choose for \( x \), you can always perform the operations required to compute \( f(x) \).

Step 3: State the Domain

Since there are no restrictions on \( x \) for the quadratic function \( f(x) = x^2 + 1 \), the domain is all real numbers.

Key Concepts

Polynomial Functions

Polynomial functions are a fundamental concept in algebra that you will come across often. A polynomial function is an expression made up of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and positive integer exponents. A simple example of a polynomial function is \( f(x) = x^2 + 1 \). Here, the term \( x^2 \) represents a variable \( x \) raised to an exponent of 2, and 1 is a constant term.

Some key characteristics of polynomial functions include:

• The degree of the polynomial is the highest power of the variable in the expression. For example, in \( f(x) = x^2 + 1 \), the degree is 2.
• Polynomial functions can be classified based on their degree: linear (degree 1), quadratic (degree 2), cubic (degree 3), and so on.
• They can have multiple terms, as seen in expressions like \( 3x^3 + 2x^2 - 5 \).
• The domain of polynomial functions is always all real numbers, unless stated otherwise.

Recognizing that polynomial functions are defined for all real numbers makes it easier to understand their applications in various mathematical contexts.

Quadratic Functions

Quadratic functions are a specific type of polynomial function where the highest power of the variable \( x \) is 2. An example of a quadratic function is \( f(x) = x^2 + 1 \).

Several features characterize quadratic functions:

• The general form is \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \).
• The graph of a quadratic function is a parabola, which can either open upwards or downwards depending on the sign of \( a \).
• Quadratic functions can intersect the x-axis at 0, 1, or 2 points based on their discriminant value.
• They always have a vertex, which is the maximum or minimum point of the parabola.

Understanding the behavior of quadratic functions, such as how they graph and their intercepts, is crucial in analyzing quadratic equations effectively.

Real Numbers

Real numbers encompass all the numbers that we can plot on a number line. They include rational numbers (both integers and fractions) and irrational numbers, which cannot be expressed as simple fractions. Some examples of real numbers are 1, -3, 0.5, and \( \pi \).

The real number system's broad definition allows polynomial and quadratic functions to operate over them without restriction. This means the domain of these functions can range over all real numbers. In the context of our exercise function \( f(x) = x^2 + 1 \), it implies:

• Every value of \( x \) from the real number set can be used in the function.
• The function is continuous and unbounded over the real number line.

Recognizing the role real numbers play in defining the domain of functions helps in a deeper understanding of how these mathematical expressions behave and are applied.