Page 10 - Pra U STPM 2022 Penggal 1 - Mathematics (T)

Mathematics Term 1 STPM Chapter 1 Functions

4. Let X = {1, 2, 3, 4,}. Show whether each of the following relations, which define the set of ordered pairs,
represent a function from X to X.
(a) f : {(2, 3), (1, 4), (2, 1), (3, 2), (4, 4)}
(b) g : {(3, 1), (4, 2), (1, 1)} 1
(c) h : {(2, 1), (3, 4), (1, 4), (2, 3), (4, 4)}
x
5. The domain of a function h, where h : x ↦ 2 , is R.
(a) Find the images of 0, 2, –1 and 6.
(b) Find the element in the domain with image 16.
(c) Find the range of the function h.

6. If f(x) = 2x – 3 and f g(x) = 2x + 1, find g(x).
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7. If f(x) = x – 1 and g f(x)= 3 + 2x – x , find g(x).
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8. If f(x) = cos x and g(x) = 1 + x, x  R, find
(a) f g(x) (b) f f(x) (c) g f(x)
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9. If f(x) =  x , x  0 and g(x) = 1 – x , x  R, find
(a) f g(x) (b) g f(x) (c) g g(x)
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10. If f : R → R and g : R → R are defined by f(x) = x + 3x + 1 and g(x) = 2x – 3, find
2
(a) f g (b) g f (c) g g (d) f f
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11. If f : R → R and g : R → R are defined by f(x) = 2x – 3 and g(x) = x + 5, find
(a) g f(2) (b) f g(3) (c) f g(a – 1)
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(d) g f(x) (e) f g(x + 1) (f) g g(x)
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Inverse functions
Let f and g be two functions defined respectively by
1
f : x ↦ 2x + 1 and g : x ↦ (x – 1)
2
1
i.e. f(x) = 2x + 1 g(x) = — (x – 1)
123
2
1423
Object of f Image of f Object of g Image of g
Then the composite function g f is defined as
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g f(x) = g[f(x)]
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= g(2x + 1)
= 1 [(2x + 1) – 1]
2
= x
i.e. we get back the object for the function f, i.e. x.
Similarly, f g(x) = f[g(x)]
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1
= f[ (x – 1)]
2
1
= 2[ (x – 1)] + 1
2
= x
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01a STPM Math T T1.indd 7 3/28/18 4:20 PM