Determine if Continuous f(x)=(x^3-8)/(x-2)
The question asks to examine whether the function f(x)=(x^3-8)/(x-2) is continuous or not. Continuity at a point means that the function is defined at that point, the limit of the function as x approaches that point exists, and the limit is equal to the function's value at that point. Since this function has a form that suggests a potential discontinuity at x=2 due to a division by zero, the problem likely requires analyzing the behavior of the function around x=2 and determining if the function can be considered continuous despite the apparent discontinuity. The question may also imply the need to use algebraic simplification to see if the function can be redefined at x=2 to resolve the discontinuity.
$f \left(\right. x \left.\right) = \frac{x^{3} - 8}{x - 2}$
Answer
Solution:
Step:1
Identify the domain to check for continuity of the function.
Step:1.1
To find the non-permissible values for $x$, equate the denominator of $\frac{x^{3} - 8}{x - 2}$ to zero: $x - 2 = 0$.
Step:1.2
Solve for $x$ by adding $2$ to both sides: $x = 2$.
Step:1.3
The domain consists of all real numbers except where the function is undefined. In interval notation, it is expressed as $(-\infty, 2) \cup (2, \infty)$. In set-builder notation, it is $\{x | x \neq 2\}$.
Step:2
Given that the domain excludes $x = 2$, the function $\frac{x^{3} - 8}{x - 2}$ is not continuous for all real numbers.
Step:3
The function is discontinuous at $x = 2$.
Knowledge Notes:
To determine if a function $f(x)$ is continuous, we need to check if it is defined for all real numbers and if there are any points where the function is not defined or has a limit that does not equal the function's value.
Domain of a Function: The set of all possible input values (usually $x$) for which the function is defined. For rational functions, values that make the denominator zero are excluded from the domain.
Continuity: A function is continuous at a point if the limit of the function as it approaches the point from both the left and the right exists and equals the function's value at that point. A function is continuous over an interval if it is continuous at every point in that interval.
Discontinuity: A function is discontinuous at a point if the function is not continuous at that point. This can occur if the function is not defined at the point, if the limits from the left and right do not exist, or if the limits exist but do not equal the function's value at the point.
Interval Notation: A way to represent a set of numbers along a line. For example, $(a, b)$ represents all numbers between $a$ and $b$, not including $a$ and $b$. $[a, b]$ includes $a$ and $b$.
Set-Builder Notation: A notation for describing a set by stating the properties that its members must satisfy. For example, $\{x | x \neq 2\}$ describes the set of all $x$ such that $x$ is not equal to $2$.
Rational Functions: Functions of the form $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$. The points where $Q(x) = 0$ are the points of discontinuity for the function.
In the given problem, the function $f(x) = \frac{x^{3} - 8}{x - 2}$ is a rational function with a denominator that becomes zero when $x = 2$. Therefore, the function is not defined at $x = 2$, which means it is discontinuous at that point and not continuous over the entire set of real numbers.
