Functions

The idea of a function is central to modern mathematics because it provides a precise way to describe how one quantity depends on another. Functions allow us to model real-world situations, from the path of a thrown ball to the growth of a population or the interest on savings. By understanding functions, we can predict outcomes, spot patterns, and solve complex problems more systematically. They also unify many areas of maths, linking algebra, geometry, and calculus through a common language of inputs and outputs..

An example of its importance is logarithms. Napier discovered/invented logs but did not understand the relationship between logs and exponentials as inverse functions - this required the precise language of functions.

Domains and Ranges, Composite and Inverse, Transformations

Why we study this

History of Functions

The idea of a function did not appear all at once. Early mathematicians, like the ancient Greeks, studied relationships between quantities using geometry, but they did not yet use our modern function language. In the 17th century, when Newton and Leibniz developed calculus, they needed a clear way to describe how one variable changes with another, such as distance with time. This pushed the function concept forward.

Function notation came later to make this idea easier to write and understand. Instead of writing long sentences, mathematicians began to use symbols like f(x) to mean “the value of the function f when the input is x”. Here, f is the rule, x is the input, and f(x) is the output. This compact notation lets us work with many different functions at once, compare them, and build more advanced ideas in algebra and calculus.

Functions Exam Tips

This might seem to be a rather abstract topic. Using notation accurately is important. You need to be aware of the difference between writing, say, f(2x),

and 2f(x), or f(x)g(x) and fg(x).

Does the equation of a circle define a function? No, because functions must map a point to another unique point, and for the circle x^2 + y^2 = 25, x = 3 is mapped to y = 4 and -4. Technically, this is an example of something called a relation. Functions are one to one or many to one, not one to many.

Domains and Ranges

The domain is the set we are mapping from, the co-domain is the set we are mapping to, and the range is the set of images. The range is a subset of the co-domain, and depends on the domain.

It is important to realise that we choose the domain - it is not 'built in' to the equation of the function. However, the domain must exclude what are called singularities. For example, for the function f(x) = 1/(x - 1) , the image of x = 1 doesn't exist - it is 'infinite', so this value must be excluded from the domain. There is a maximal domain for this function, namely all real numbers except 1, but this is not necessarily the domain, though it is often taken to be the 'default' domain.

If you are asked to find the range of a function with a given domain, the best way is to sketch the function, taking account of the domain. Here is an example. [Text]

Composite and inverse functions

It's easy to get the order of composite functions wrong: fg(x) means, rather confusingly, do g followed by f. So if f(x) = x^2 and g(x) = 2x + 1, then fg(x) = f([g(x)]) = f(2x + 1) = (2x + 1)^2, whereas gf(x) = g([f(x)]) = g(x^2) = 2x^2 + 1. Notice they are not the same! I find putting in the square bracket (shown in red) helps to trigger the correct procedure.

With inverse functions, the best way is to write y = f(x), then swap x and y round, and solve for y.

However, inverse functions only exist if the domain of f is one-to-one (not many-to-one). For example for f(x) = x^2, if the domain is all real numbers then f(2) = f(-2) = 4, so it is many to one (viz both 2 and -2 are mapped to the same value, 4). But if the domain is restricted to positive real numbers, then the function becomes one to one (as -2 is no longer in the domain). The inverse function is then the square root function. This restriction in the domain can be seen in the trigonometric functions sin x and cos x. The inverse functions arcsin(x) and arccos(x) are reflections of these domains in the line y = x.

f(x) = sin(x)

sin(x) and arcsin x

f(x) = cos(x)

cos(x) and arccos(x)

Transformations

Knowing the relationship between f(x) and f(x) + k, f(x + k), kf(x) and f(kx) and their respective graphs gives us a powerful tool for sketching the graphs of these related functions. We can also sketch other related functions by treating them as a sequence of transformations.

Let's look at the transformations related to various functions:

f(x) + k. This corresponds to a translation of k in the y-direction.

k f(x) This corresponds to a one-way stretch in the y-direction.

f(x + k). This corresponds to the translation of -k in the x-direction.

f(kx). This is a one-way stretch in the x-direction.

Here is an example of a composite of two transformations.[text]

Notice that when the k is outside the brackets, it affect the y-coordinate. When the 'k' is inside the bracket. the transformation affects the x-coordinate.

Three special cases are worth knowing. When k = -1 in kf(x) we have -f(x) - this is the reflection of f(x) in the x-axis.

If f(-x) = f(x), f is called an even function, and is symmetrical about the y-axis. Examples are x^2 and cos(x).

When k = -1 in f(kx), we get f(-x), which is the reflection of f(x) in the y-axis.

If f(-x) = -f(x), f is called an odd function, and has point symmetry about the origin. Examples are x^3 and sin(x).

And the modulus of f(x) reflects any points below the x-axis in that axis - see below.

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