Question

Given that f(x)=

==
3r-1
a. Graph f(x).
b. Identify the vertical asymptote.
c. Identify the horizontal asymptote.
d. Describe the end behavior

Answer 1

a. A graph of the function
`f(x) = (3x-1)/(x-4)` with its vertical and horizontal asymptotes is shown in the image below.

b. The vertical asymptote is x = 4.

c. The horizontal asymptote is y = 3.

d. The end behavior is that as x approaches negative infinity (-∞), f(x) approaches negative infinity (-∞).

In Mathematics and Euclidean Geometry, a rational function is a type of function which is expressed as a fraction that is composed of two main parts and these include the following:

Numerator

Denominator

Based on the information provided above, we can logically deduce the following rational function;

`f(x) = (3x-1)/(x-4)`

Part b

In order to graph any rational function, you should determine the values for which it is undefined. This ultimately implies that, a function is considered as undefined when the value of the denominator is equal to zero, which represents vertical asymptote lines;

x - 4 = 0

x = 0 + 4

x = 4 (vertical asymptote)

Part c.

Horizontal asymptote: y = 3 (since the degree of the denominator is the same as that of the numerator).

Part d.

Based on the graph shown below, the end behavior is that as x approaches negative infinity (-∞), f(x) approaches negative infinity (-∞);

As x → -∞, f(x) → -∞