Question

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

Short answer

The domain of the function is all real numbers.

Step-by-step solution

- Identify the Function Type

The given function is a polynomial function, specifically a quadratic function, because it is written in the form of a polynomial where the highest power of x is 2.

- Understand the Properties of Polynomial Functions

Polynomial functions, including quadratic functions, are defined for all real numbers because there are no restrictions such as division by zero or taking the square root of a negative number.

- State the Domain

Since polynomial functions have no restrictions on their input values, the domain of the function is all real numbers.

Key concepts

Polynomial Functions

Polynomial functions are expressions that involve sums and products of variables and coefficients. These functions can have terms with variables raised to whole number exponents.
For example, a polynomial function can look like this: \(f(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 10\).
Each term in this function is made up of a coefficient (like 3, -2, 5, etc.) and a variable raised to an exponent (like \(x^4, x^3, x^2\), and so on.
Now, an important feature of polynomial functions is that they are defined for all real numbers. This means you can plug any real number into the function and get a valid output without running into mathematical issues like division by zero or taking the square root of a negative number.

Quadratic Functions

A quadratic function is a specific type of polynomial function where the highest exponent of the variable is 2.
It generally takes the form: \(f(x) = ax^2 + bx + c\).
For our given function \(f(x) = x^2 + 2\), this is a quadratic function where \(a = 1\), \(b = 0\), and \(c = 2\).
One key aspect of quadratic functions is that their graphs form parabolas, which can open upwards or downwards depending on the sign of 'a'.
For example, if 'a' is positive, the parabola opens upwards. If 'a' is negative, it opens downwards.
Just like other polynomial functions, quadratic functions are also defined for all real numbers.

Real Numbers

Real numbers include all the numbers that can appear on the number line.
This set encompasses all the positive and negative integers, rational numbers (fractions and decimals that terminate or repeat), and irrational numbers (decimals that do not terminate or repeat, like \(\text{π}\) and \( \text{√2}\)).
Because polynomial functions, including quadratic ones, are designed to accept any real number as an input, the domain of these functions is all real numbers.
This is why we can confidently say that the function \(f(x) = x^2 + 2\) is defined for any real number, and thus the domain of \(f(x)\) is all real numbers.