Determine if Continuous f(x)=(5+x)/(x-2)
The given problem "Determine if Continuous f(x)=(5+x)/(x-2)" asks to check whether the given function \( f(x) = \frac{5+x}{x-2} \) is continuous across its entire domain. Specifically, it is asking you to analyze the function's behavior at all points in its domain to see if there are any points of discontinuity, such as jumps, holes, or vertical asymptotes, and to determine the overall continuity of the function.
$f \left(\right. x \left.\right) = \frac{5 + x}{x - 2}$
Answer
Solution:
Determine the Continuity of the Function $f(x) = \frac{5 + x}{x - 2}$
Step 1: Identify the domain where the function is defined.
Step 1.1: To find the domain, set the denominator of $\frac{5 + x}{x - 2}$ to zero and solve for $x$. $$x - 2 = 0$$ Step 1.2: Solve the equation by adding $2$ to both sides.
$$x = 2$$ Step 1.3: The function is defined for all $x$ except where the denominator is zero. The domain is expressed as:
Interval Notation: $(-\infty, 2) \cup (2, \infty)$ Set-Builder Notation: $\{ x | x \neq 2 \}$ Step 2: The function is not continuous for all real numbers since it is undefined at $x = 2$.
Step 3: There is no further action required as the continuity has been assessed.
Knowledge Notes:
To determine if a function $f(x)$ is continuous, we need to ensure that it is defined for all values within its domain. A function is continuous at a point $x = a$ if the following conditions are met:
$f(a)$ is defined.
The limit of $f(x)$ as $x$ approaches $a$ exists.
The limit of $f(x)$ as $x$ approaches $a$ is equal to $f(a)$.
For rational functions, such as $f(x) = \frac{5 + x}{x - 2}$, discontinuities can occur where the denominator is zero since division by zero is undefined. To find the domain of a rational function, we set the denominator equal to zero and solve for $x$. The solutions to this equation are the points of discontinuity.
Interval notation is a way of writing subsets of the real number line. An interval notation for a domain typically looks like $(a, b)$, $[a, b]$, $(-\infty, a)$, or $(b, \infty)$, where parentheses indicate that the endpoints are not included, and square brackets indicate they are included.
Set-builder notation is another way to describe a set of numbers, written as $\{ x | \text{property of } x \}$, which reads as "the set of all $x$ such that $x$ has the given property."
In this case, since the function $f(x) = \frac{5 + x}{x - 2}$ is undefined at $x = 2$, it is not continuous at that point, and thus not continuous over the entire real number line.
