analyze the graph of the function
R(x)=x^2-12/x-6
a) What is the domain of R(x)?
b) What is the equation of the vertical asymptotes of R(x)?
c) What is the equation of the horizontal or oblique asymptote of R(x)?
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(x^2-12)/(x-6)
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Let f (×)= b×
To analyze the graph of the function R(x) = (x^2 - 12) / (x - 6), we can follow these steps:
a) Determine the domain of R(x):
The domain of a function is the set of all possible input values that the function can accept. In this case, the function R(x) is defined for all real numbers except any value that makes the denominator equal to zero. So, we need to find out when x - 6 = 0. Solving for x, we find that x = 6. Therefore, the domain of R(x) is all real numbers except x = 6.
b) Find the equation(s) of the vertical asymptotes of R(x):
Vertical asymptotes occur when there are values of x that make the denominator (x - 6) equal to zero. We found that x = 6 is the value that makes the denominator zero. So, the equation of the vertical asymptote is x = 6.
c) Determine the equation of the horizontal or oblique asymptote:
To find the horizontal or oblique asymptote, we need to evaluate the behavior of the function as x approaches positive and negative infinity. We can do this by looking at the degrees of the numerator and denominator.
In this case, the degree of the numerator is 2 and the degree of the denominator is 1. Since the numerator has a higher degree, there is no horizontal asymptote.
However, we can still check for an oblique asymptote. To do this, we perform long division to divide the numerator (x^2 - 12) by the denominator (x - 6). The quotient we get will represent the equation of the oblique asymptote, if it exists.
Performing the long division, we get:
(x^2 - 12) ÷ (x - 6) = (x + 6)
The quotient is x + 6, which represents the equation of the oblique asymptote.
In summary:
a) The domain of R(x) is all real numbers except x = 6.
b) The equation of the vertical asymptote of R(x) is x = 6.
c) There is no horizontal asymptote, but there is an oblique asymptote given by the equation y = x + 6.