Find the Asymptotes (3x-8)/(x-7)
The question is asking for the identification of the asymptotes of the function given by the expression (3x - 8) / (x - 7). An asymptote refers to a line that a graph of a function approaches but never actually reaches. There are typically two types of asymptotes to consider for rational functions like this one - vertical asymptotes where the denominator of the fraction is zero (causing undefined values for the function) and horizontal or oblique asymptotes that describe the behavior of the graph as the input (x) gets very large in the positive or negative direction. The request is to determine these lines, if any, for the given function.
$\frac{3 x - 8}{x - 7}$
Answer
Solution:
Step 1:
Identify the values of $x$ that cause the function $f(x) = \frac{3x - 8}{x - 7}$ to be undefined. This occurs when the denominator is zero: $x - 7 = 0$.
Step 2:
Examine the general form of a rational function $f(x) = \frac{ax^n}{bx^m}$, where $n$ represents the degree of the polynomial in the numerator and $m$ represents the degree of the polynomial in the denominator. The rules for determining horizontal asymptotes are as follows:
If $n < m$, the horizontal asymptote is the x-axis, or $y = 0$.
If $n = m$, the horizontal asymptote is the line $y = \frac{a}{b}$.
If $n > m$, there is no horizontal asymptote, but there may be an oblique asymptote.
Step 3:
Determine the values of $n$ and $m$ for our function. In this case, $n = 1$ and $m = 1$.
Step 4:
Since $n$ is equal to $m$, the horizontal asymptote is found using the formula $y = \frac{a}{b}$. For our function, $a = 3$ and $b = 1$, so 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: $x = 7$
- Horizontal Asymptote: $y = 3$
- No Oblique Asymptote
Step 7:
The process is complete, and all asymptotes have been identified.
Knowledge Notes:
To find the asymptotes of a rational function, one must understand the behavior of the function as it approaches certain critical values. Here are the relevant knowledge points:
Undefined Points: A rational function becomes undefined where its denominator is zero. The values of $x$ that make the denominator zero are the locations of vertical asymptotes.
Degrees of Polynomials: The degree of a polynomial is the highest power of $x$ with a non-zero coefficient. In the context of rational functions, the degrees of the numerator and denominator polynomials are used to determine the presence and equation of horizontal asymptotes.
Horizontal Asymptotes: These are horizontal lines that the graph of the 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 polynomials ($n$ 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 (Slant) Asymptotes: These occur when the degree of the numerator is exactly one more than the degree of the denominator. The equation of the oblique asymptote can be found by performing polynomial long division.
Vertical Asymptotes: These occur at the values of $x$ that make the denominator zero (and are not canceled by the numerator). The graph of the function will approach infinity or negative infinity near these values.
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