Cambridge International AS and A Level Mathematics Pure Mathematics 1

Functions

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P1^

4

(i) Temperature measured in Celsius  temperature measured in Fahrenheit.

(ii) Marks in an examination  grade awarded.

(iii) Distance measured in light years  distance measured in metres.

(iv) Number of stops travelled on the London Underground  fare.

You can decide whether an algebraic mapping is a function, and whether it has

an inverse function, by looking at its graph. The curve or line representing a one-

to-one function does not double back on itself and has no turning points. The x

values cover the full domain and the y values give the range. Figure 4.5 illustrates

the functions f, g and h given on the previous page.

Now look at f(x) = x^2 for x ∈  (figure 4.6). You can see that there are two

distinct input values giving the same output: for example f(2) = f(−2) = 4. When

you want to reverse the effect of the function, you have a mapping which for a

single input of 4 gives two outputs, −2 and +2. Such a mapping is not a function.

You can make a new function, g(x) = x^2 by restricting the domain to + (the set

of positive real numbers). This is shown in figure 4.7. The function g(x) is a

one-to-one function and its inverse is given by g−^1 (x) = (^) x since the sign
means ‘the positive square root of’.
y
O x
y y
O x O x
–1
1
y = f(x) y = g(x) y = h(x)
Figure 4.5
f(x) f(x) = x^2
4
–2 O 2 x
Figure 4.6