Cambridge International AS and A Level Mathematics Pure Mathematics 1

Functions

P1^

4

The full definition of the inverse function

is therefore:

f−^1 (x) = x−^2 for x  2.

The function and its inverse function are

shown in figure 4.11.

(ii) f(7) = 72 + 2 = 51

f−^1 f(7) = f−^1 (51) = 51 −=^27

Applying the function followed by its inverse brings you back to the original

input value.

Note

Part (ii) of Example 4.6 illustrates an important general result. For any function f(x)

with an inverse f−^1 (x), f−^1 f(x) = x. Similarly ff−^1 (x) = x. The effects of a function and its

inverse can be thought of as cancelling each other out.

ExERCISE 4B  1  The functions f, g and h are defined for x ∈ by f(x) = x^3 , g(x) = 2 x and

h(x) = x + 2. Find each of the following, in terms of x.

(i) fg (ii) gf (iii) fh (iv) hf (v) fgh

(vi) ghf (vii) g^2 (viii) (fh)^2 (ix) h^2

2  Find the inverses of the following functions.

(i) f(x) = 2 x + 7, x ∈ (ii) f(x) = 4 − x, x ∈

(iii) f(x) = (^) 2–^4 x, x ≠ 2 (iv) f(x) = x^2 − 3, x  0
3  The function f is defined by f(x) = (x − 2)^2 + 3 for x  2.
(i) Sketch the graph of f(x).
(ii) On the same axes, sketch the graph of f−^1 (x) without finding its equation.
4  Express the following in terms of the functions f: x  x and g: x  x + 4 for
x  0.
(i) x  x+^4 (ii) x  x + 8
(iii) x  x+^8 (iv) x  x+^4
5  A function f is defined by:
f: x 

1

x^ x^ ∈^ , x^ ≠ 0.

Find (i) f^2 (x) (ii) f^3 (x) (iii) f−^1 (x) (iv) f^999 (x).

y

O

x

f–1 (x) = x – 2 for x ≥ 2.

y = f(x)

y = f–1(x)

y = x

Figure 4.11