Identify the holes, vertical asymptotes, x-intercepts, horizontal asymptote, domain, and range of:
x^2-2x-8
f(x)= ---------
-2x+2
Please explain how you got all of this info.
To identify the holes, vertical asymptotes, x-intercepts, horizontal asymptote, domain, and range of the given function:
Step 1: Identify the Holes
To find any holes in the function, we need to check if there are any common factors in both the numerator and denominator that can be canceled out. In this case, there are no common factors, so there are no holes in the function.
Step 2: Find Vertical Asymptotes
To find the vertical asymptotes, we need to solve the equation -2x + 2 = 0 for x. Solving this equation, we get:
-2x = -2
x = 1
Thus, the vertical asymptote for the function is x = 1.
Step 3: Find X-intercepts
To find the x-intercepts, we need to set the numerator equal to zero and solve for x. Solving the equation x^2 - 2x - 8 = 0, we can factor this equation as:
(x - 4)(x + 2) = 0
Setting each factor equal to zero, we get:
x - 4 = 0 --> x = 4
x + 2 = 0 --> x = -2
So, the x-intercepts of the function are x = 4 and x = -2.
Step 4: Find Horizontal Asymptote
To find the horizontal asymptote, we examine the degree of the numerator and denominator. Since the degree of the numerator (x^2 - 2x - 8) is equal to the degree of the denominator (-2x + 2), there is a horizontal asymptote.
In this case, when the degrees are equal, we look at the ratio of the leading coefficients of the two polynomials. The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is -2. Thus, the horizontal asymptote is y = 1 / -2, which simplifies to y = -1/2.
Step 5: Determine the Domain
The domain of a function is the set of all possible x-values for which the function is defined. In this case, the only restriction on the domain is that x ≠ 1 (since it would make the denominator zero, causing division by zero). Therefore, the domain is all real numbers except for x = 1.
Step 6: Determine the Range
The range of a function is the set of all possible y-values that the function can produce. In this case, since there is a horizontal asymptote at y = -1/2, and the function is continuous and has no holes, the range is all real numbers except for y = -1/2.
In summary, for the given function f(x) = (x^2 - 2x - 8) / (-2x + 2):
Holes: None
Vertical Asymptotes: x = 1
X-intercepts: x = 4 and x = -2
Horizontal Asymptote: y = -1/2
Domain: All real numbers except x = 1
Range: All real numbers except y = -1/2
To identify the holes, vertical asymptotes, x-intercepts, horizontal asymptote, domain, and range of a rational function, you need to analyze the equation and simplify it.
First, let's identify any potential holes in the function. Holes occur when there is a common factor in both the numerator and denominator, leading to a cancellation. In this case, we don't have any common factors between the numerator and denominator, so there are no holes.
Next, let's find the vertical asymptotes. Vertical asymptotes occur when the denominator is equal to zero, causing the function to approach infinity or negative infinity. To find the vertical asymptotes, we need to solve the equation -2x + 2 = 0 for x:
-2x + 2 = 0
-2x = -2
x = 1
So, we have a vertical asymptote at x = 1.
To find the x-intercepts, we need to solve the equation f(x) = 0. Set the numerator equal to zero and solve:
x^2 - 2x - 8 = 0
This equation can be factored as (x - 4)(x + 2) = 0. Thus, x = 4 and x = -2 are the x-intercepts.
Now, let's determine the horizontal asymptote. To find the horizontal asymptote, we need to compare 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 degree of the numerator is greater, there is no horizontal asymptote. Instead, the function will have an oblique (slant) asymptote. To find it, perform polynomial long division of the numerator by the denominator:
x
--------
-2x + 2 | x^2 - 2x - 8
-x^2 + x^2
---------------------
-3x - 8
The result of polynomial long division is -3x - 8. Therefore, the slant asymptote is y = -3x - 8.
Regarding the domain and range, the domain is the set of all possible x-values that the function can take. In this case, the only restriction is that x cannot be equal to 1 (due to the vertical asymptote). So, the domain is all real numbers except x = 1.
The range is the set of all possible y-values that the function can take. Since the function has a slant asymptote and no horizontal asymptote, the range is all real numbers.
To summarize:
- Holes: None
- Vertical asymptotes: x = 1
- X-intercepts: x = 4, x = -2
- Horizontal asymptote: None (slant asymptote y = -3x - 8)
- Domain: All real numbers except x = 1
- Range: All real numbers