Page 7 - Pra U STPM 2022 Penggal 1 - Mathematics (T)

Mathematics Term 1 STPM Chapter 1 Functions

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(b) y = x + 3x + 2
For each value of x  R, the value of y is unique.
1 Thus y is a function of x.
When x = 0, y = 0 + 0 + 2 = 2.
When x = –3, y = 9 – 9 + 2 = 2.
There exist two values of x  R with the same image.
Thus the function is not one-to-one.
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(c) y = x – 2
y = ± x – 2
For each value of x  R with x . 2, there exist two possible values of y.
For example, y = ±2 when x = 6.
Thus y is not a function of x.

Algebraic operations on functions

Algebraic operations can be applied to functions.
If f : x ↦ f(x) and g : x ↦ g(x), then
(i) f + g : x ↦ f(x) + g(x),
(ii) f – g : x ↦ f(x) – g(x),
(iii) f · g : x ↦ f(x) · g(x),
(iv) f : x ↦ f(x) , g(x) ≠ 0,
g g(x)
(v) af : x ↦ af(x), a  R.

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For example, let f(x) = x + 1 and g(x) = x , where the domain is the set of real numbers, R.
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Then, f + g : x ↦ (x + 1) + x , x  R
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f – g : x ↦ (x + 1) – x , x  R
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f · g : x ↦ (x + 1)x , x  R
f : x ↦ x + 1 , x ≠ 0
g x 2
2f : x ↦ 2(x + 1), x  R

Composite functions
Suppose that the functions f and g are defined for the set of real numbers R, with
f : x ↦ y and g : y ↦ z
This means that we can create a new function which maps x directly to z. This function is called a composite
function and is written as g f, where g f(x) = g[f(x)].
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Consider the diagram below, which shows the mapping of f, g and g f.
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01a STPM Math T T1.indd 4 3/28/18 4:20 PM