This is part of our
GRE Math Essentials project is the best complement to our
Official GRE Quant Book. It provides a cutting-edge, in-depth overview of all the math concepts to achieve 170 in the Quantitative Reasoning portion of the GRE. The book still remains our hallmark. However, the following chapters will give you a lot of tips, tricks, and shortcuts to make your quant preparation more robust and solid.
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 FunctionsThe 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 \(x^2 + 6x + 9\)
In general \(gf(x)\) is not equal to \(fg(x)\)
source:
https://www.mcckc.edu/https://www.mathcentre.ac.uk/Attachment:
function.jpg [ 9.92 KiB | Viewed 40481 times ]