m1a functions

8/15/2019 M1a Functions

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Functions 1

Kelvin Soh

14th January 2014

Functions and their domains

A function, f , is a rule which assigns to each number, x, in its domainexactly one number, f (x). It is often useful to view a function as a machine.It takes in an input x and, based on the input, gives an output, f (x).

Picture credit: http://thelivingpearl.com

For example, for the equation y = x2, y is a function of x. This is becausegiven any value of x, (e.g. -2), our equation tells us what y is with noambiguity (e.g. 4). Thus, we can write y = f (x) = x2.However, x in general is not a function of y. This is because if someone tellsus a y value (e.g. 4), we may not be able to tell exactly what x is (e.g. xmay be 2 or -2).

The domain of f , usually denoted Df is the set of inputs we wish to consider.This is important because our rule (i.e. function) may not work for somevalues. For example, f (x) = 1

x does not make sense for x = 0 because we

cannot divide by 0. Thus, a possible domain will be the set (−∞, 0)∪(0,∞).

There are also other reasons for considering the domain. For example, x mayhave a physical meaning (e.g. weight of an apple). In that case, it does not

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make sense to talk about negative values of x. Thus, it may be useful to

restrict the domain of a function of x to the set [0,∞).

Remark 1: Outside of this topic, domains are often omitted and assumedto be the biggest possible set for which the function is valid. For problemsums, the domain may further be restricted such that values obtained makessense (e.g. the weight example from the previous paragraph).

Remark 2: For this topic, every function that is defined should come witha description of the domain.

In our syllabus, a function is defined in the following fashion:

f : x → x + 1, 0 ≤ x


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Example 1 (Finding range of a function with a graph)

Find the range of the following function:

f : x → x2 + 1, −1 < x ≤ 2.

x

y

y = f (x)

−2 −1 1 2 3

1

2

3

4

5

0

Df

Rf

Note that the graph of the function is drawn with a solid line because of thedomain. The dotted line represents the graph of y = x2 + 1 for all values of x and is only for reference.

The domain, (−1, 2] can be seen as the x-values the graph take. Meanwhile,the range corresponds to the y-values the graph takes.

Thus, reading off the graph, Rf = [1, 5].

Remark 4: The end points, as well as the maximum and minimum points,are of crucial importance when finding the range of a function. We can findthe coordinates for stationary points by differentiation or from our G.C. if exact values are not required. For quadratic functions, completing the squareis another method to obtain the maximum/minimum point.

References

1) T. B. Ng, Calculus: An Introduction (Springer, 1997).2) http://thelivingpearl.com.

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Basic Practice

1. The function f is defined as follows:

f : x → x2 − 4x + 5, 0 < x

m1a functions

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