1. Definition.

A function of a real variable x with domain D is a rule that assigns a unique real number to each number x in D. Functions are often given letter names such as f, g, F, or φ. We often call x the independent variable or the argument of f. If g is a function and x is a number in D, then g(x) denotes the number that g assigns to x. We sometimes make the idea that F has an argument (we substitute a number for the variable in F) explicit by writing F(·). In the case of two variables we sometimes use y = f(x) for the value of f evaluated at the number x. Note the difference between φ and φ(x)

2. The domain of a function.

The domain is the set of all values that can be substituted for x in the function f(·). If a function f is defined using an algebraic formula, we normally adopt the convention that the domain consists of all values of the independent variable for which the function gives a meaningful value (unless the domain is explicitly mentioned).

3. The range of a function.

Let g be a function with domain D. The set of all values g(x) that the function assumes is called the range of g. To show that a number, say a, is in the range of a function f, we must find a number x such that f(x) = a. Here are some example functions for which to find the domain and the range.

1: \(f(x) = x, 0 ≤ x ≤ 60\)

2: \(g(x) = \frac{x^2}{20}, 0 ≤ x ≤ 60\)

3: \(h(x) = \frac{12}{x^2}\)

4: \(φ(x) = 3x^2\)

5: \(x = −\frac{1}{2} y^2, y ≥ 0\)

6: \(f(x) = \frac{3x}{x^2 − 4}\)

7: \(g(x) = \sqrt{4 − 3x}\)


4. The graph of a function.

When the rule that defines a function f is given by an equation in y and x, the graph of f is the graph of the equation, that is the set of points (x, y) in the xy-plane that satisfies the equation. Another way to say this is that the graph of the function g is the set of all point (x, g(x)), where x belongs to the domain of g.

5. The vertical line test.

As set of points in the xy-plane is the graph of a function if and only if a vertical line intersects the graph in at most one point.



Compound Functions


The composition of two functions g and f is the new function we get by performing f first, and then performing g. For example, if we let f be the function given by \(f(x) = x^2\) and let g be the function given by \(g(x) = x + 3\), then the composition of g with f is called gf and is worked out as

\(gf(x) = g(f(x))\)

So we write down what f(x) is first, and then we apply g to the whole of f(x). In this case, if we apply g to something we add 3 to it. So if we apply g to \(x^2\), we add three to \(x^2\). So we obtain

\(gf(x) = g(f(x)) = g(x^2) = x^2 + 3 \)

The order in which we compose functions makes a big difference to the result. You can see this if we change the order of the functions in the first example. We have taken \(f(x) = x^2\) and \(g(x) = x + 3\). Then fg(x) is given by taking g(x), which is x + 3, and applying f to all of it. This gives us

\(fg(x) = f(x + 3) = (x + 3)^2 = x^2 + 6x + 9 \)

You can see that this is not the same as gf(x), because

\(gf(x) = x^2 + 3\)

and this does not in general equal \(x^2 + 6x + 9\)



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