Determine if Continuous f(x)=8/x
The question asks to evaluate whether a given function, specifically f(x) = 8/x, is continuous. Continuity, in a mathematical sense, generally means that there are no breaks, jumps, or holes in the graph of the function. It implies that for every point x within the function's domain, the limit of the function as it approaches x exists and is equal to the function's value at x. For this specific function, you would analyze its continuity by looking at its domain and at any points where the function might fail to be continuous.
$f \left(\right. x \left.\right) = \frac{8}{x}$
Answer
Solution:
Step:1
Identify the domain to check the continuity of the function.
Step:1.1
To find where the function is not defined, equate the denominator of $\frac{8}{x}$ to $0$. We get $x = 0$.
Step:1.2
The domain consists of all $x$ values for which the function is defined. In Interval Notation, this is: $\left( -\infty , 0 \right) \cup \left( 0 , \infty \right)$. In Set-Builder Notation, it is: $\{ x | x \neq 0 \}$.
Step:2
The function $\frac{8}{x}$ does not have a domain that includes all real numbers, indicating that it is not continuous across the entire set of real numbers. Therefore, the function is not continuous.
Step:3
Knowledge Notes:
To determine the continuity of a function like $f(x) = \frac{8}{x}$, we need to understand the concept of continuity and the domain of a function.
Continuity: A function is continuous at a point if the limit of the function as it approaches the point from both sides is equal to the function's value at that point. A function is continuous over an interval if it is continuous at every point in that interval.
Domain: The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, such as $f(x) = \frac{8}{x}$, the domain excludes values that would make the denominator zero, since division by zero is undefined.
Interval Notation: This is a way of writing subsets of the real number line. An interval notation like $(a, b)$ represents all numbers between $a$ and $b$, not including $a$ and $b$ themselves. The notation $[a, b]$ includes the endpoints $a$ and $b$.
Set-Builder Notation: This notation describes a set by specifying a property that its members must satisfy. For example, $\{ x | x \neq 0 \}$ describes all real numbers $x$ such that $x$ is not equal to zero.
Limits and Continuity: To be continuous at a point, a function must be defined at that point, the limit as it approaches the point must exist, and the limit must equal the function's value at that point.
In the case of $f(x) = \frac{8}{x}$, the function is not defined at $x = 0$, which means it is not continuous at $x = 0$. The domain of $f(x)$, therefore, is all real numbers except zero, and the function is discontinuous at $x = 0$.
