Find the Domain h(x)=3x-2
The problem is asking to determine the set of all possible input values x (real numbers) for the function h(x) = 3x - 2 that will result in real number outputs for h(x). This set of input values is known as the domain of the function.
$h \left(\right. x \left.\right) = 3 x - 2$
Answer
Solution:
Determine the Domain of h(x) = 3x - 2
Step 1:
Identify the set of all possible x-values for which h(x) is defined. For the function h(x) = 3x - 2, there are no restrictions such as division by zero or square roots of negative numbers. Therefore, the domain includes all real numbers.
Interval Notation: $(-\infty, \infty)$
Set-Builder Notation: $\{x | x \in \mathbb{R}\}$
Step 2:
Since there are no further restrictions or steps to consider, the domain has been fully determined in Step 1.
Knowledge Notes:
The domain of a function is the set of all possible input values (usually x-values) for which the function is defined. When determining the domain, we look for values that might cause the function to be undefined, such as:
Division by zero, which is undefined in real numbers.
Taking the square root (or any even root) of a negative number, which is not a real number.
For linear functions like h(x) = 3x - 2, which is a polynomial of degree one, there are no such restrictions. Polynomials are defined for all real numbers, so the domain of a linear function is always all real numbers, denoted by $\mathbb{R}$.
Interval notation is a way of writing subsets of the real number line. An interval that includes all real numbers is written as $(-\infty, \infty)$, where the parentheses indicate that infinity is not a real number and is not included in the set.
Set-builder notation is another way to describe a set, using a rule or condition. For the domain of a function that includes all real numbers, the set-builder notation is $\{x | x \in \mathbb{R}\}$, which reads as "the set of all x such that x is a real number."
