Suppose that f(x)=
(5x)
_________
(x^2)−x−6
a. Determine the domain of f
b. Determine the vertical asymptote(s), if any, of the graph of f
c. Determine the horizontal asymptote(s), if any, of the graph of f
d, Sketch the graph of f
To determine the domain of a function, you need to find all the values of x for which the function is defined. In this case, we need to find the values of x that make the denominator (x^2 - x - 6) nonzero since division by zero is undefined.
a. Determine the domain of f:
First, let's factor the denominator:
(x^2 - x - 6) can be factored as (x - 3)(x + 2)
Now, set each factor equal to zero and solve for x:
(x - 3) = 0 => x = 3
(x + 2) = 0 => x = -2
So, the function is undefined at x = 3 and x = -2. Therefore, the domain of f is all real numbers except x = 3 and x = -2. In interval notation, the domain is (-∞, -2) U (-2, 3) U (3, ∞).
b. Determine the vertical asymptote(s), if any, of the graph of f:
To find the vertical asymptote(s), we need to check for any values of x where the function approaches infinity or negative infinity as x approaches that value.
In this case, since the function is a ratio of two polynomials, the vertical asymptote(s) occur where the denominator is zero. From factoring the denominator earlier, we found x = 3 and x = -2 as points where the denominator is zero. Therefore, the vertical asymptotes are x = 3 and x = -2.
c. Determine the horizontal asymptote(s), if any, of the graph of f:
To find the horizontal asymptote(s), we need to check the behavior of the function as x approaches positive or negative infinity.
In this case, the degree of the numerator (1st degree) is less than the degree of the denominator (2nd degree). Therefore, the horizontal asymptote is y = 0.
d. Sketch the graph of f:
To sketch the graph of f, plot the vertical asymptotes at x = -2 and x = 3, and the horizontal asymptote y = 0. Also, plot a few additional points to get an idea of the shape of the graph. You can choose some x-values and compute their corresponding y-values using the given function f(x).
Keep in mind that this sketch is just a rough idea and the actual graph may have more details and features.