Find the Asymptotes f(x)=(5+x)/(5-x)
The question asks for the identification of the asymptotes of the function f(x) = (5 + x) / (5 - x). An asymptote of a function is a line such that the distance between the curve and the line approaches zero as they tend to infinity. There are typically two types of asymptotes - vertical and horizontal (or oblique). Vertical asymptotes occur where the function is undefined, typically resulting from values that make the denominator zero. Horizontal or oblique asymptotes describe the behavior of the function as x approaches infinity or negative infinity. The question specifically requires determining where these asymptotes are for the given rational function.
$f \left(\right. x \left.\right) = \frac{5 + x}{5 - x}$
Answer
Solution:
Step 1:
Determine the values of $x$ for which $\frac{5 + x}{5 - x}$ does not exist. This occurs when $x = 5$.
Step 2:
Examine the general form of a rational function $R(x) = \frac{a x^{n}}{b x^{m}}$. The degrees of the polynomial in the numerator and denominator ($n$ and $m$, respectively) dictate the horizontal asymptote as follows:
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 3:
Identify the degrees of the numerator and denominator. For our function, $n = 1$ and $m = 1$.
Step 4:
Since the degrees are equal ($n = m$), calculate the horizontal asymptote using the coefficients $a$ and $b$. Here, $a = 1$ and $b = -1$, so the horizontal asymptote is $y = \frac{1}{-1}$, which simplifies to $y = -1$.
Step 5:
Check for an oblique asymptote. Since the degree of the numerator is not greater than the degree of the denominator, there is no oblique asymptote.
Step 6:
Compile the list of 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 must first understand what an asymptote is. An asymptote is a line that the graph of a function approaches but never touches. There are three types of asymptotes: vertical, horizontal, and oblique.
Vertical Asymptotes occur at values of $x$ where the function is undefined, often due to division by zero. To find them, set the denominator equal to zero and solve for $x$.
Horizontal Asymptotes are found by comparing the degrees of the numerator and denominator ($n$ and $m$). If the degree of the numerator is less than the degree of the denominator ($n < m$), the horizontal asymptote is $y = 0$. If the degrees are equal ($n = m$), the horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator, respectively. If the degree of the numerator is greater than the degree of the denominator ($n > m$), there is no horizontal asymptote.
Oblique Asymptotes may occur when the degree of the numerator is exactly one more than the degree of the denominator ($n = m + 1$). In such cases, you can find the oblique asymptote by performing polynomial long division.
Rational Functions are functions represented by the ratio of two polynomials. The general form is $R(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials.
Degrees of Polynomials are determined by the highest power of $x$ in the polynomial. The degree influences the end behavior of the function and the presence of horizontal or oblique asymptotes.
Coefficients are the numerical factors in front of variables in polynomials. The leading coefficient is the coefficient of the term with the highest power of $x$.
