Page 18 - Pra U STPM 2022 Penggal 1 - Mathematics (T)

Mathematics Term 1 STPM Chapter 1 Functions
Graphs of rational functions

A rational function is a function in the form f(x) , where f(x) and g(x) are algebraic functions with g(x) ≠ 0.
g(x) 1
1
An example of a rational function is xy = 1, i.e. y = 1 or x = .
x y
Notice that when x → ±∞, y → 0 and when y → ± ∞, x → 0. The lines y = 0 and x = 0 are known as the
asymptotes for the curve xy = 1.

In general, a curve which approaches a straight line when x → ± ∞ or when y → ±∞ will have the straight
line as its asymptote.

y

xy = 1

x
O
asymptotes

Figure 1.13

Example 9

Sketch the curve with equation y = 2x – 1 .
x + 1
State the asymptotes of the curve.

Solution: y = 2x – 1
x + 1
1
x(2 – — )
Rewriting the equation as y = x ,
1
x(1 + — )
x
1
2 – —
x
We get y = .
1
1 + —
x
When x → ± ∞, 1 → 0 and y → 2.
x
Hence, y = 2 is an asymptote to the curve y = 2x – 1 .
x + 1
When x = –1, y = 2x – 1 is undefined.
x + 1
Hence, x = –1 is also an asymptote to the curve.
When x = 0, y = –1.
When y = 0, x = 1 .
2

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