Find the Asymptotes f(x)=(x-1)/(x^2-4x+3)
This problem is asking for the asymptotes of the function f(x) = (x-1)/(x^2-4x+3). An asymptote is a line that the graph of a function approaches but never touches. There are two types of asymptotes that may be relevant here: horizontal and vertical. A horizontal asymptote would be a horizontal line that the graph of the function approaches as x goes to infinity or negative infinity. A vertical asymptote is a vertical line near which the function goes to infinity (or minus infinity) very quickly, typically as x approaches a certain value.
For rational functions like this one, vertical asymptotes can be found where the denominator is zero (since division by zero is undefined), but only if the numerator isn't also zero at those points (which would be a removable discontinuity instead). The horizontal asymptote can be found by comparing the degrees of the numerator and denominator polynomials and applying the appropriate rules based on these degrees.
In summary, the question is asking to determine where, if anywhere, the graph of the given rational function f(x) is approaching some line(s) as it stretches infinitely in the horizontal direction (left/right) and near specific points in the vertical direction (up/down).
$f \left(\right. x \left.\right) = \frac{x - 1}{x^{2} - 4 x + 3}$
Answer
Solution:
Step 1:
Identify the values for which the function $\frac{x - 1}{x^{2} - 4x + 3}$ is not defined. These values are $x = 1$ and $x = 3$.
Step 2:
Examine the behavior of the function as $x$ approaches $3$. As $x$ approaches $3$ from the left, $\frac{x - 1}{x^{2} - 4x + 3} \rightarrow -\infty$. As $x$ approaches $3$ from the right, $\frac{x - 1}{x^{2} - 4x + 3} \rightarrow \infty$. Hence, $x = 3$ is a vertical asymptote.
Step 3:
Review the general rules for horizontal asymptotes in a rational function $R(x) = \frac{ax^n}{bx^m}$:
If the degree of the numerator $n$ is less than the degree of the denominator $m$, the horizontal asymptote is the x-axis, $y = 0$.
If $n$ equals $m$, the horizontal asymptote is the line $y = \frac{a}{b}$.
If $n$ is greater than $m$, there is no horizontal asymptote; instead, there may be an oblique asymptote.
Step 4:
Determine the degrees $n$ and $m$ of the numerator and denominator, respectively. In this case, $n = 1$ and $m = 2$.
Step 5:
Since the degree of the numerator $n$ is less than the degree of the denominator $m$, the horizontal asymptote is the x-axis, $y = 0$.
Step 6:
Conclude that there is no oblique asymptote, as the degree of the numerator is not greater than the degree of the denominator.
Step 7:
Summarize the asymptotes of the function:
- Vertical Asymptotes: $x = 3$
- Horizontal Asymptotes: $y = 0$
- No Oblique Asymptotes
Knowledge Notes:
Vertical Asymptotes: These occur at values of $x$ where the function becomes undefined, typically where the denominator of a rational function is zero.
Horizontal Asymptotes: These are horizontal lines that the graph of a function approaches as $x$ goes to infinity or negative infinity. The rules for finding horizontal asymptotes depend on the degrees of the numerator and denominator of the rational function.
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 are found by performing polynomial long division.
Rational Functions: A function of the form $R(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.
Behavior of Functions: As $x$ approaches certain critical values, the behavior of the function can indicate the presence of asymptotes. This behavior can include approaching infinity, negative infinity, or a constant value.
Degrees of Polynomials: The degree of a polynomial is the highest power of $x$ with a non-zero coefficient. In a rational function, comparing the degrees of the numerator and denominator helps determine the presence and type of asymptotes.
