Determine if Continuous f(x)=(x^2-x-2)/(x+1)
The question asks to determine whether the function f(x) = (x^2 - x - 2)/(x + 1) is continuous. Basically, the problem requires you to investigate the behavior of that rational function and establish if there are any points at which the function is not continuous, such as points where the function is undefined or has a discontinuity. The task may involve looking at the domain of the function, identifying any problematic points (like where the denominator is zero), and using the definition of continuity to analyze the function around those points.
$f \left(\right. x \left.\right) = \frac{x^{2} - x - 2}{x + 1}$
Answer
Solution:
Step 1: Identify the Domain for Continuity
To assess continuity, we must first establish the domain where the function is defined.
Step 1.1: Locate Points of Discontinuity
Solve for $x$ when the denominator of $\frac{x^2 - x - 2}{x + 1}$ is zero, since division by zero is undefined. $$x + 1 = 0$$
Step 1.2: Solve for the Discontinuity
Isolate $x$ by subtracting $1$ from both sides of the equation to find the point of discontinuity.
$$x = -1$$
Step 1.3: Define the Domain
The domain consists of all $x$ values where the function is defined, which excludes the point of discontinuity.
Interval Notation: $(-\infty, -1) \cup (-1, \infty)$ Set-Builder Notation: $\{x | x \neq -1\}$
Step 2: Determine Continuity Over the Domain
Given that the domain excludes $x = -1$, the function $\frac{x^2 - x - 2}{x + 1}$ is not continuous over the entire set of real numbers.
Knowledge Notes:
To determine if a function is continuous, we need to consider the following:
Domain of a Function: The set of all possible input values (usually $x$ values) for which the function is defined. A function is continuous if it is defined and smooth (without breaks, holes, or jumps) over its entire domain.
Discontinuity: A function is discontinuous at a point if it is not continuous at that point. Types of discontinuities include:
Point Discontinuity: When a function is not defined at a point, but is defined immediately on either side of that point.
Jump Discontinuity: When a function has a sudden jump in value at a point.
Infinite Discontinuity: When a function approaches infinity near a point.
Rational Functions: A function of the form $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials. The function is undefined when $Q(x) = 0$. The points where $Q(x) = 0$ are the potential points of discontinuity.
Interval Notation: A way to describe the domain or range of a function using intervals. For example, $(-\infty, -1) \cup (-1, \infty)$ indicates all real numbers except $-1$.
Set-Builder Notation: Another way to describe a set, using a rule to define the members of the set. For example, $\{x | x \neq -1\}$ describes all real numbers $x$ such that $x$ is not equal to $-1$.
In the given problem, the rational function $\frac{x^2 - x - 2}{x + 1}$ is undefined at $x = -1$ because the denominator equals zero at this point. Therefore, the function has a point discontinuity at $x = -1$ and is not continuous over the entire set of real numbers.
