Page 13 - Pra U STPM 2022 Penggal 1 - Mathematics (T)

Mathematics Term 1 STPM Chapter 1 Functions

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2. Given that g : x ↦ x + x + 1 and x  X, where X = {–2, –1, 0, 1, 2}.
(a) Sketch a diagram to represent g.
(b) State the domain of g.
1 (c) State the range Y of g.
(d) Is g a one-to-one function?
3. Given that h : x ↦ 2x – 3, where x  {1, 3, 5, 7, 9} and that the range is {–5, –3, –1, 3, 5, 7, 11, 15}.
(a) Sketch a diagram to represent h.
(b) Find h(1) and h(9).
(c) Is h a one-to-one function?

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4. The function f is defined by f : x ↦ x – x, x  R. State whether f is one-to-one, giving a reason for the
statement. If the domain of f is restricted to the subset of R for which x  k, find the least value of k for
which f is one-to-one.
5. The function f is defined by
f : x ↦ x , x  R
2
x + 1
1
If a  R and a ≠ 0, find the image of — under f.

a
Deduce that f is not one-to-one.
Show that if a, b  R with a . b  1, then f(b) . f(a).
Deduce that, if the domain of f is restricted to the subset of R given by {x : x  1}, then f is one-to-one.
State the range of f in this case.

6. Find the inverse of each of the following functions, stating its domain.
(a) f : x ↦ x – 2, x  R
2
(b) f : x ↦ x + 1, x  0
2
(c) f : x ↦ (x – 1) , x  1
(d) g : x ↦ (x – 2)(x – 4), x  3
+
(e) g : x ↦ (x – 3)(x + 3), x  R
(f) h : x ↦ 2 , x ≠ 3
x – 3
(g) h : x ↦ x + 2 , x ≠ 2
x – 2
2
7. Given that f(x) = x + 1 and x  2, find f (x).
–1
2x + 1
2
Prove that (f f)(x) = x.
–1
°
8. The functions f and g are defined for x  R (excluding –1, 0 and 1) by
f : x ↦ 1 + x and g : x ↦ 1
1 – x x
Show that (f g) = f g.
–1
°
°
9. Sketch the graph of each of the following functions and its inverse.
(a) f(x) = x + 3, x  R
(b) f(x) = 2x – 1, x  R
(c) f(x) = (x – 1)(x + 1), x  R
+
2
(d) f(x) = x + 3x – 4, x . – 3
2

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01a STPM Math T T1.indd 10 3/28/18 4:20 PM