Question

F(x)= 2/(x^2-2x-3)

find and show work:
1. domain and range
2. x and y intercept
3. horizontal asymptote
4. vertical asymptote

Answer 1

The function f(x) = 2/(x^2 - 2x - 3) has a domain of all real numbers except x ≠ 3 and x ≠ -1, and a range of all real numbers. There are no x-intercepts; the y-intercept is at (0, -2/3). It has a horizontal asymptote at y = 0 and vertical asymptotes at x = 3 and x = -1.

Step-by-step explanation:

The function you are analyzing is f(x) = 2/(x^2 - 2x - 3). Let's find the details one by one:

1. Domain: The domain of a function consists of all the possible input values (x-values) that will not result in an undefined expression. For this rational function, we must consider that division by zero is undefined. The denominator, x^2 - 2x - 3, factors to (x - 3)(x + 1). Thus, the domain is x ≠ 3 and x ≠ -1 because these values would make the denominator equal to zero. In interval notation, the domain is (-∞, -1) ∪ (-1, 3) ∪ (3, ∞).
2. Range: The range is a bit more complex to find. Looking at the function's behavior, as x approaches the vertical asymptotes or ∞, the function approaches 0. Therefore, the range should include all real numbers except a possible value that f(x) might never take, which in this case there are none. Hence, the range is all real numbers, or in interval notation, (-∞, ∞).
3. X-intercept: To find the x-intercept, set the output f(x) equal to 0 and solve for x. Since 2/(x^2 - 2x -3) can never be 0 (the numerator is constant and non-zero), there are no x-intercepts.
4. Y-intercept: To find the y-intercept, evaluate the function at x = 0, resulting in f(0) = 2/(-3), which simplifies to -2/3. Hence, the y-intercept is at (0, -2/3).
5. Horizontal asymptote: If the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator, the horizontal asymptote is at y = 0. Since this is the case with our function, there is a horizontal asymptote at y = 0.
6. Vertical asymptotes: Vertical asymptotes occur where the function is undefined, which is at the values that make the denominator zero. Therefore, from the domain, we can see that the vertical asymptotes are at x = 3 and x = -1.