Determine if Continuous f(x)=3/x
The problem you've presented is to analyze a given function, \( f(x) = \frac{3}{x} \), and determine if it is continuous. To assess the continuity of this function, you would consider whether the function has any breaks, holes, jumps, or vertical asymptotes over its domain. In other words, the question is asking you to check if the function is well-defined and smoothly varying for all values in its domain, without any interruptions.
$f \left(\right. x \left.\right) = \frac{3}{x}$
Answer
Solution:
Step 1: Identify the Domain for Continuity
To assess the continuity of the function $f(x) = \frac{3}{x}$, we first need to determine its domain.
Step 1.1: Locate Points of Discontinuity
To find where the function is not defined, we solve for $x$ when the denominator is zero: $x = 0$.
Step 1.2: Define the Domain
The function is defined for all $x$ except where $x = 0$. The domain in interval notation is: $(-\infty, 0) \cup (0, \infty)$. In set-builder notation, it is: $\{x | x \neq 0\}$.
Step 2: Determine Continuity Over the Domain
Given that the domain excludes $x = 0$, the function $f(x) = \frac{3}{x}$ cannot be continuous over the entire set of real numbers.
Step 3: Conclusion
The function is not continuous across all real numbers due to the discontinuity at $x = 0$.
Knowledge Notes:
To determine if a function $f(x)$ is continuous, we must examine its domain and identify any points of discontinuity. A function is continuous at a point if the following three conditions are met:
The function is defined at the point.
The limit of the function as it approaches the point exists.
The limit of the function as it approaches the point is equal to the function's value at that point.
For rational functions, such as $f(x) = \frac{3}{x}$, discontinuities can occur where the denominator is zero since division by zero is undefined. The domain of a function is the set of all possible input values (x-values) for which the function is defined.
Interval notation is a way of writing subsets of the real number line. An interval that does not include its endpoints is denoted with parentheses, e.g., $(a, b)$, whereas an interval that includes its endpoints is denoted with square brackets, e.g., $[a, b]$.
Set-builder notation is another way to describe a set by specifying a property that its members must satisfy. For example, $\{x | x \neq 0\}$ describes the set of all real numbers $x$ such that $x$ is not equal to zero.
In this case, the function $f(x) = \frac{3}{x}$ is undefined at $x = 0$, which creates a point of discontinuity. Therefore, the function is not continuous over the entire real number line. The domain of $f(x)$, excluding the point of discontinuity, is all real numbers except zero, which can be expressed in interval notation as $(-\infty, 0) \cup (0, \infty)$ and in set-builder notation as $\{x | x \neq 0\}$.
