Find the Asymptotes f(x)=x square root of 7-x
The question asks to determine the asymptotes of the function f(x) = x√(7-x). This typically involves analyzing the behavior of the function as the variable x approaches certain critical values where the function might not be defined or where it tends to infinity. Specifically, the question might be looking for vertical asymptotes, which are x-values where the function approaches infinity as x approaches them, and horizontal or oblique asymptotes, which describe the behavior of the function as x goes to positive or negative infinity. To address this question, one would generally investigate the limits of f(x) as x approaches specific values or infinity, and apply knowledge of limits and asymptotic behavior to identify such asymptotes.
$f \left(\right. x \left.\right) = x \sqrt{7 - x}$
Answer
Solution:
Step 1:
Determine the domain where the function $f(x) = x\sqrt{7 - x}$ is not defined, which is when $x > 7$.
Step 2:
Identify any vertical asymptotes, which occur at points of infinite discontinuity. The function does not have any vertical asymptotes.
Step 3:
For a rational function $R(x) = \frac{a x^{n}}{b x^{m}}$, where $n$ is the degree of the numerator and $m$ is the degree of the denominator:
If $n < m$, the horizontal asymptote is $y = 0$.
If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$.
If $n > m$, there are no horizontal asymptotes, but there may be an oblique asymptote.
Step 4:
Since the function $f(x)$ has a degree of $1$ in the numerator and no denominator, it does not have a horizontal asymptote.
Step 5:
Attempt to find any oblique asymptotes using polynomial division. However, due to the presence of a radical in the expression, polynomial division is not applicable.
Step 6:
Compile the list of asymptotes for the function:
- No Vertical Asymptotes
- No Horizontal Asymptotes
- Cannot Determine Oblique Asymptotes
Knowledge Notes:
When analyzing functions for asymptotes, it's important to understand the different types of asymptotes and how they relate to the function's behavior:
Vertical Asymptotes: These occur at values of $x$ where the function approaches infinity. They are typically found at points where the denominator of a rational function is zero, but the numerator is not zero.
Horizontal Asymptotes: These are horizontal lines that the graph of the function approaches as $x$ goes to infinity or negative infinity. For rational functions, horizontal asymptotes are determined by comparing the degrees of the numerator and the denominator, as described in Step 3.
Oblique Asymptotes: Also known as slant asymptotes, these occur when the degree of the numerator is exactly one more than the degree of the denominator. They can be found using polynomial long division, but this method does not apply to functions with radicals or other non-polynomial expressions.
Domain of a Function: The set of all possible $x$-values for which the function is defined. For the given function $f(x) = x\sqrt{7 - x}$, the domain is restricted by the square root, which requires the argument to be non-negative.
Polynomial Division: A method used to divide polynomials, which can also be used to find oblique asymptotes for rational functions. However, it is not applicable to functions that are not polynomials, such as those containing radicals.
Continuity and Discontinuity: A function is continuous at a point if it is defined there, and the limit as $x$ approaches that point exists and equals the function's value at that point. A discontinuity is a point where the function is not continuous. Vertical asymptotes are associated with infinite discontinuities.
