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
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.
x |
0 | 2 | 3 | 3.5 |
f(x) | -0.5 | -1 | -2 | -4 |
Let us now evaluate f at values of x close to 4 such that x > 4
x | 6 | 5 | 4.5 | 4.6 |
f(x) | 1 | 2 | 4 | 0.33 |
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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