Question

Graph the rational function : f(x)=x-4/x^2-16. Graph the rational function. Identify the key features 1) Domain 2) V.A. 3) H.A. 4) Slant

Answer 1

A graph of the function
`f(x) = (x-4)/(x^2-16)` with its vertical and horizontal asymptotes is shown in the image below. The key features of the graph are;

1. Domain: (-∞, -4) ∪ (-4, 4) ∪ (4, ∞).

2. Vertical asymptote (V.A): x = -4

3. Horizontal asymptote (H.A): y = 0

4. Slant: -x - y = 4

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) = (x-4)/(x^2-16)`

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^(2) -16=0\\\\x=\pm√(16) \\\\`

x = -4 (vertical asymptote)

Domain: (-∞, -4) ∪ (-4, 4) ∪ (4, ∞).

Horizontal asymptote: y = 0 (since the degree of the denominator is greater than that of the numerator).