Find the Asymptotes f(x)=(3x-7)/(x-6)
The problem is asking to determine the asymptotes of the given function, which is a rational function represented as f(x)=(3x-7)/(x-6). The Asymptotes of a function are lines that the graph of the function approaches as the input (x) either goes to infinity or to a specific point where the function is undefined. There are two types of asymptotes to consider for a rational function: vertical and horizontal (or oblique/slant asymptotes if the degree of the numerator is exactly one higher than the denominator).
For the given function, you would look for:
Vertical asymptotes by finding the values of x for which the denominator becomes zero, as long as they do not also make the numerator zero.
Horizontal asymptotes by comparing the degrees of the numerator and the denominator.
Or if applicable, an oblique (slant) asymptote, if the degree of the numerator is exactly one more than the degree of the denominator.
The problem requires an analysis of the function to determine where the function cannot exist due to division by zero and where it approaches but never reaches along the x and y axes as x approaches infinity or negative infinity.
$f \left(\right. x \left.\right) = \frac{3 x - 7}{x - 6}$
Answer
Solution:
Step 1:
Identify the values of $x$ that make the function $\frac{3x - 7}{x - 6}$ undefined. This occurs when the denominator equals zero: $x - 6 = 0$, hence $x = 6$.
Step 2:
Examine the general form of a rational function $R(x) = \frac{ax^n}{bx^m}$, where $n$ is the degree of the polynomial in the numerator and $m$ is the degree of the polynomial in the denominator.
If $n < m$, the horizontal asymptote is the x-axis, given by $y = 0$.
If $n = m$, the horizontal asymptote is the line $y = \frac{a}{b}$.
If $n > m$, there is no horizontal asymptote; instead, there may be an oblique asymptote.
Step 3:
Determine the values of $n$ and $m$ for the given function. In this case, $n = 1$ and $m = 1$.
Step 4:
Since $n$ is equal to $m$, the horizontal asymptote can be found using the formula $y = \frac{a}{b}$, where $a = 3$ and $b = 1$. Therefore, the horizontal asymptote is $y = 3$.
Step 5:
An oblique asymptote is not present because the degree of the numerator is not greater than the degree of the denominator.
Step 6:
Compile the list of asymptotes for the function:
- Vertical Asymptote: $x = 6$
- Horizontal Asymptote: $y = 3$
- There are no Oblique Asymptotes.
Step 7:
The process is complete, and the asymptotes have been identified.
Knowledge Notes:
To find the asymptotes of a rational function, one must understand the behavior of the function as $x$ approaches certain critical values. Asymptotes are lines that the graph of a function approaches but never touches.
Vertical Asymptotes: These occur at values of $x$ that make the denominator of a rational function zero (provided that the numerator is not also zero at those points). To find them, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes: These are found by comparing the degrees of the numerator and the denominator ($n$ and $m$, respectively).
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 (or slant) asymptote. This is found by performing polynomial long division or synthetic division.
Rational Functions: A rational function is a ratio of two polynomials. It is written in the form $R(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials.
Understanding these concepts is crucial for analyzing the behavior of rational functions and identifying their asymptotes.
