Question

Find the domain of the function. \(g(x)=1-2 x^{2}\)

Short answer

The domain of the function \(g(x)=1-2 x^{2}\) is all real numbers, or in interval notation, (-∞, ∞).

Step-by-step solution

Recognize the type of function

Recognize that the function \(g(x)=1-2 x^{2}\) is a polynomial. This is because it is an expression consisting of variables and coefficients, which involves the operations of addition, subtraction, multiplication, and non-negative integer exponents.

Analyze the implications on the domain

For most simple functions, including polynomial functions like the one given, any real number can be plugged in for the variable. There are no restrictions on the domain since there are no values that we aren’t allowed to use (values that would cause us to divide by zero or take the square root of a negative). The function can therefore accept any real number as input.

State the domain

Since the function can accept any real number as input, the domain of the function is 'all real numbers' or in interval notation, it is expressed as (-∞, ∞), which means that x is greater than negative infinity and less than positive infinity.

Key concepts

Domain of a Function

Understanding the domain of a function is crucial when analyzing any mathematical expression. The domain represents all the possible input values (often represented by 'x') that a function can accept without causing any undefined scenarios. In simpler terms, it tells us what values we are allowed to put into the function. These scenarios can include division by zero or computing the square root of a negative number, which are not defined in the set of real numbers. However, for polynomial functions like the one discussed, which is given as \( g(x) = 1 - 2x^2 \), determining the domain is straightforward. Polynomial functions are defined for all real numbers because they don't have operations that restrict the input, such as division by zero. Thus, the domain of the function \( g(x) \) is all real numbers.

Real Numbers

The set of real numbers includes all the numbers that can be found on the number line. This set is vast; it encompasses both rational numbers, like integers and fractions, and irrational numbers, which are numbers that cannot be expressed as simple fractions. Real numbers are critical in mathematics because they represent quantities that we deal with in real life, like distances and temperatures. We use them all the time in functions, especially in polynomials like \( g(x) = 1 - 2x^2 \), where any real number can be an input. This is because polynomial functions do not have any restrictions like square roots of negative numbers or divisions by zero that can make a number outside the domain of real numbers.

Interval Notation

Interval notation is a convenient way of writing down the domain or range of a function. It provides a clear method to express an interval of numbers by using brackets and parentheses to denote endpoints. If you see \((a, b)\), it means all numbers between \(a\) and \(b\) but not including \(a\) and \(b\) themselves. Meanwhile, \([a, b]\) would include both \(a\) and \(b\). When saying a function is defined for all real numbers, we often use the notation \((-\infty, \infty)\). This means that there is no beginning or end limit for the input values. In the context of our function \(g(x) = 1 - 2x^2\), the domain of all real numbers is written as \((-\infty, \infty)\), showing that the function accepts any real number as an input.