4.5 Analysis and graphing of Rational Functions:

Let us understand this part of rational functions using an example as given below:

Analyze and graph the rational function whose rule is

f (x) = 2x - 3

x – 4

Vertical Asymptotes

Let,

f(x) = 2

x - 4

The domain of f is the set of all real numbers except 4, since 4 makes the denominator zero and the division by zero is not defined in mathematics. Let us calculate function f at values of x close to 4 such that x < 4.

Let us now evaluate f at values of x close to 4 such that x > 4

1. i) As x approaches 4 from the left or by values smaller than 4, f (x) decreases without bound.
2. ii) As x approaches 4 from the right or by values larger than 4, f (x) increases without bound.

We say that the line x = 3, broken line, is the vertical asymptote for the graph of f.

In general, the line x = a is a vertical asymptote for the graph of f if f (x) either increases or decreases without bound as x approaches a from the right or from the left. This is symbolically written as:

f(x) → ∞ f(x) → -∞

as x → a+ or x → a-

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