For each of the rational functions find: a. domain b. holes c. vertical asymptotes d. horizontal asymptotes e. oblique asymptotes f. y-intercept g. x-intercepts
1. f(x)= x^2+x-2 / x^2-x-6
Y = (x^2+x-2)/(x^2-x-6)
Y = (x+2)(x-1)/(x+2)(x-3) = (x-1)/(x-3)
a. Domain = All real values of X except
3. Note: 3 gives a denominator of zero
which makes the fraction undefined.
f. When Y = 0,(x-1) = 0
x-1 = 0, X = 1.
X-Intercept = 1.
To find the domain of a rational function, we need to determine the values of x for which the function is defined. In other words, we need to find the values that make the denominator non-zero. For the function f(x) = (x^2 + x - 2) / (x^2 - x - 6), we have a quadratic in both the numerator and the denominator.
a. Domain: To find the domain, we need to determine the values of x that make the denominator (x^2 - x - 6) non-zero. We can factor the denominator as (x-3)(x+2). So the function is defined for all values of x except when x = 3 and x = -2. Therefore, the domain is all real numbers except x = 3 and x = -2.
b. Holes: To find any holes in the rational function, we need to simplify the function and check if any factors cancel out. In this case, we can simplify f(x) as f(x) = (x - 1) / (x - 3). We canceled out the common factor of (x + 2) from the numerator and denominator. Since the factor (x - 3) cancels out, we have a hole at x = 3.
c. Vertical Asymptotes: To find the vertical asymptotes, we need to determine the values of x for which the denominator (x^2 - x - 6) becomes zero. We can factor the denominator as (x-3)(x+2). Therefore, we have vertical asymptotes at x = 3 and x = -2.
d. Horizontal Asymptotes: To find the horizontal asymptotes, we look at the degrees of the numerator and denominator. In this case, both the numerator and denominator have the same degree, which is 2. Therefore, we look at the leading coefficients. For f(x), the leading coefficient in both the numerator and denominator is 1. So the horizontal asymptote is y = 1.
e. Oblique Asymptotes: To find oblique asymptotes, we check the degrees of the numerator and denominator. If the degree of the numerator is exactly one more than the degree of the denominator, we can find an oblique asymptote. However, in this case, the degrees of the numerator and denominator are the same (both 2). Therefore, there are no oblique asymptotes.
f. y-intercept: To find the y-intercept, we substitute x = 0 into the function. For f(x) = (x^2 + x - 2)/(x^2 - x - 6), when x = 0, we get f(0) = -2 / -6 = 1/3. Therefore, the y-intercept is (0, 1/3).
g. x-intercepts: To find the x-intercepts, we set the numerator equal to zero and solve for x. For f(x) = (x^2 + x - 2)/(x^2 - x - 6), we set x^2 + x - 2 = 0. Factoring the quadratic, we get (x-1)(x+2) = 0. So the x-intercepts are x = 1 and x = -2.
To find the domain of the rational function, we look for values of x for which the denominator is not equal to zero. So, let's first find the values of x that make the denominator zero.
1. Find the domain:
Solving the denominator, x^2 - x - 6 = 0, we can factor it as (x - 3)(x + 2) = 0. This gives us two solutions, x = 3 and x = -2. Therefore, the domain of the rational function is all real numbers except x = 3 and x = -2.
Domain: All real numbers except x = 3 and x = -2.
2. Find the holes:
To find the holes, we need to determine if there are any common factors between the numerator and the denominator that can be canceled out. Simplifying the function, we have:
f(x) = (x + 2)(x - 1) / (x - 3)(x + 2)
Notice that (x + 2) appears in both the numerator and denominator. So, we can cancel it out:
f(x) = (x - 1) / (x - 3)
Therefore, there is a hole at x = -2.
Holes: x = -2.
3. Find the vertical asymptotes:
Vertical asymptotes occur when the denominator is equal to zero, but the numerator is not zero. From our earlier calculation, we found that the denominator is equal to zero at x = 3. By looking at the function, we can see that the numerator is not zero at x = 3. Therefore, we have a vertical asymptote at x = 3.
Vertical asymptotes: x = 3.
4. Find the horizontal asymptotes:
To find horizontal asymptotes, we need to compare the degrees of the numerator and the denominator. The degree of the numerator is 2 and the degree of the denominator is also 2.
Since the degrees are the same, we need to compare the leading coefficients. The leading coefficient of the numerator is 1 and the leading coefficient of the denominator is also 1.
Therefore, the rational function has a horizontal asymptote at y = (leading coefficient of the numerator) / (leading coefficient of the denominator).
Horizontal asymptote: y = 1/1 = 1.
5. Find the oblique asymptote:
To determine if there is an oblique asymptote, we need to check if the degree of the numerator is exactly one more than the degree of the denominator.
In this case, the degree of the numerator (2) is not exactly one more than the degree of the denominator (2). Therefore, there is no oblique asymptote.
Oblique asymptote: Non-existent.
6. Find the y-intercept:
The y-intercept occurs when x = 0. Plugging in x = 0 into the function:
f(0) = (0^2 + 0 - 2) / (0^2 - 0 - 6) = -2 / -6 = 1/3
Therefore, the y-intercept is (0, 1/3).
Y-intercept: (0, 1/3).
7. Find the x-intercepts:
The x-intercepts occur when the numerator is equal to zero. Setting the numerator equal to zero and factoring it, we have:
x^2 + x - 2 = 0
(x + 2)(x - 1) = 0
Solving for x, we get two x-intercepts:
x + 2 = 0 -> x = -2
x - 1 = 0 -> x = 1
Therefore, the x-intercepts are at (-2, 0) and (1, 0).
X-intercepts: (-2, 0) and (1, 0).