Find the Asymptotes f(x)=(x^2+4x-45)/(x^2-25)
The question is asking for the determination of the asymptotes of the given function $f(x) = (x^2 + 4x - 45) / (x^2 - 25)$. An asymptote is a line that a graph approaches as it goes to infinity or negative infinity. For rational functions like the one given, there can be vertical and horizontal (or oblique) asymptotes. A vertical asymptote occurs at values of x that make the denominator equal to zero (unless the same factor is also in the numerator, in which case the point is a removable discontinuity). A horizontal or oblique asymptote describes the behavior of the graph as x goes to infinity or negative infinity and is determined by the degrees and leading coefficients of the polynomial in the numerator and the polynomial in the denominator. The problem is essentially asking for the identification and calculation of these lines for the specific function provided.
$f \left(\right. x \left.\right) = \frac{x^{2} + 4 x - 45}{x^{2} - 25}$
Answer
Solution:
Step 1:
Identify the values of $x$ that cause the function $\frac{x^{2} + 4x - 45}{x^{2} - 25}$ to be undefined. These values are $x = -5$ and $x = 5$.
Step 2:
Examine the behavior of the function as $x$ approaches $-5$. The function tends toward $-\infty$ when approaching from the left and $\infty$ when approaching from the right. Therefore, $x = -5$ is a vertical asymptote.
Step 3:
Analyze the degrees of the numerator and denominator for the rational function $\frac{ax^{n}}{bx^{m}}$. The horizontal asymptote depends on the relationship between $n$ and $m$:
If $n < m$, the horizontal asymptote is $y = 0$.
If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$.
If $n > m$, there is no horizontal asymptote, but there may be an oblique asymptote.
Step 4:
Determine the degrees of the numerator and denominator. For our function, both $n$ and $m$ are equal to 2.
Step 5:
Since the degrees of the numerator and denominator are equal ($n = m$), the horizontal asymptote is given by $y = \frac{a}{b}$. In this case, $a = 1$ and $b = 1$, so the horizontal asymptote is $y = 1$.
Step 6:
Conclude that there are no oblique asymptotes because the degree of the numerator is not greater than the degree of the denominator.
Step 7:
Compile the list of all asymptotes for the function:
- Vertical Asymptotes: $x = -5$
- Horizontal Asymptotes: $y = 1$
- No Oblique Asymptotes
Knowledge Notes:
To find the asymptotes of a rational function, we follow these steps:
Vertical Asymptotes: These occur at the values of $x$ that make the denominator zero (as long as those values do not also make the numerator zero). To find them, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes: These depend on the degrees of the numerator ($n$) and the denominator ($m$) of the rational function $\frac{ax^{n}}{bx^{m}}$:
If $n < m$, the horizontal asymptote is $y = 0$.
If $n = m$, the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively.
If $n > m$, there is no horizontal asymptote.
Oblique Asymptotes: If the degree of the numerator is exactly one more than the degree of the denominator ($n = m + 1$), the function may have an oblique (slant) asymptote. To find it, perform polynomial long division or synthetic division.
Behavior Near Vertical Asymptotes: To determine how the function behaves near a vertical asymptote, investigate the limits of the function as $x$ approaches the asymptote from both the left and the right.
Undefined Points: It's important to note that if a value of $x$ makes both the numerator and the denominator zero, it is a point of indeterminate form and not a vertical asymptote. Further investigation is needed to determine the behavior at that point.
Simplifying the Function: Sometimes, the original function can be simplified by factoring and canceling out common terms in the numerator and the denominator. This can change the set of vertical asymptotes, so it should be done before finding them.
