Problem 12 - Intermediate Algebra

Question

Specify the domain for each of the functions. $$f(x)=x^{2}+1$$

Step-by-Step Solution

Answer

The domain is all real numbers, \((- fty, fty)\).

Step 1: Identify the Function

The given function is \( f(x) = x^2 + 1 \). This is a polynomial function of degree 2.

Step 2: Recognize the Function Type

Since \( f(x) = x^2 + 1 \) is a polynomial function, it is composed of terms added together, each of which is the product of a constant and a non-negative integer power of \( x \).

Step 3: Determine the Domain of Polynomial Functions

The domain of a polynomial function, like \( f(x) = x^2 + 1 \), is all real numbers. Polynomial functions have a domain of \((-fty, fty)\) because they do not have any restrictions such as division by zero or square roots of negative numbers.

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression consisting of variables, coefficients, and non-negative integer exponents summed together. A simple example is the function \[f(x) = x^2 + 1\]. Here, the term \(x^2\) is the variable raised to an exponent and the number 1 is a constant. Each part, like \(x^2\) and +1, is called a term.

• Polynomial functions can have one or more terms.
• The degree of a polynomial is determined by the highest exponent of the variable. For example, in \(x^2 + 1\), the degree is 2.
• They are defined for all real number inputs, meaning we can plug in any real number for \(x\).

The structure of polynomial functions allows them to handle any real value without the risk of undefined operations like division by zero.

Real Numbers

Real numbers include all rational and irrational numbers, forming a complete set for expressing values on a continuous number line. From integers to fractions, and from repeating decimals to non-repeating decimals, they all fall under the category of real numbers, making them essential in various mathematical functions.
Here are some examples:

• Positive and negative integers such as -3, 0, 4.
• Fractions like \(\frac{1}{2}\).
• Non-terminating decimals such as \(\pi\) (approximately 3.14159...)

Real numbers can seamlessly be used in polynomial functions such as \(f(x) = x^2 + 1\), providing an unrestricted set of inputs (the domain) for functions.

Function Domain

The domain of a function is the complete set of all possible input values (usually \(x\)-values) that a function can accept. In simpler words, it’s all the possible values you can use in a function without causing any mathematical mishaps.

Determining Domain in Polynomial Functions

Polynomial functions like \(f(x) = x^2 + 1\) are very forgiving when it comes to domains. Because they involve only non-negative integer exponents and are made up of algebraic sums, they don't impose any restrictions due to operations like square roots of negatives or division by zero.

• The domain of any polynomial function is all real numbers, noted as \((\-\infty, \infty)\), meaning you can use any real number as input without any problem.
• This characteristic makes polynomials simpler to work with in comparison to other types of functions that might be more restrictive.

As a result, you don't have to worry about finding values that might lead toward undefined results or constraints within polynomial functions.