The concept of a function

When a quantity \(y\) is uniquely figured out by some various other quantity \(x\) together a result of some dominion or formula, then we say the \(y\) is a function of \(x\). (In other words, for each worth of \(x\), over there is at many one corresponding value that \(y\).)

We begin with six examples in i m sorry both \(x\) and also \(y\) are real numbers:

• \(y=x+2\) • \(y=3x^2-7\) • \(y=\sin x\) • \(y=2^x\) • \(y=\dfrac1x\) • \(y=\log_2x\).

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We draw their graphs in the normal way, v the \(x\)-axis horizontal and also the \(y\)-axis vertical.

in-depth description

The an initial four functions are comparable in that their recipe "work" for all real numbers \(x\). Because that \(y=\dfrac1x\), we plainly need \(x\neq 0\), and also for \(y=\log_2x\), we need \(x>0\). Us will comment on this further in the ar Domains and ranges.

What is a relation?

There are numerous naturally developing formulas who graphs are not the graphs the functions. For example:


The very first graph is a circle, the second is one ellipse, the third is two straight lines, and the fourth is a hyperbola. In each example, there space values that \(x\) for which there space two values of \(y\). Therefore these room not graphs the functions.

It turns out the the most useful concept to assist describe and understand this worry is really general.


A relation top top the genuine numbers is any kind of subset the \(\mathbbR\times\mathbbR\). That is, a relation on the reals is a set of ordered bag of genuine numbers.

Thus the 4 graphs above and the graphs of the six instance functions are all connections on the real numbers. Indeed, the graph the any function is a relation. Official speaking, a function is a relation together that, for each \(x\), there is at most one bespeak pair \((x,y)\).


The heat \(y=x\) divides the number aircraft into 3 relations.

The relationship \(R_1 = \\, (x,y) \mid x=y \,\\) is the heat itself. Keep in mind that, for each \(x\), over there is only one worth of \(y\).

The relationship \(R_2 = \{\, (x,y) \mid x The relation \(R_3 = \\, (x,y) \mid x>y \,\\) consists of all points strictly below the heat \(y=x\).

example The equation the a line \(l\) in the airplane is given by \(ax+by+c=0\). This line determines, in a organic way, three connections on the reals: \beginalign* R_1 &= \\, (x,y) \mid ax+by+c=0 \,\ \\ R_2 &= \\, (x,y) \mid ax+by+c0 \,\. \endalign*

Graphs and also the vertical-line test

We have seen the graphs of numerous naturally described functions. A wise question come ask, because that a given graph in \(\mathbbR^2\), is whether it is the graph that a function.

The vertical-line test gives a basic geometric test because that answering this question:

If we can draw a vertical heat \(x=c\) that cut the graph more than once, climate the graph is no the graph of a function.

Returning to the graph of \(x^2+y^2=25\), we watch that the vertical line \(x=3\) meets the graph in ~ both \((3,4)\) and also \((3,-4)\).

thorough description

Hence, the graph of \(x^2+y^2=25\) is no the graph of a function. The line \(x=6\) walk not satisfy the graph at all, but this does no matter. In general: for \(-5 because that \(c=-5\) and for \(c=5\), the line \(x=c\) meets the graph when because that \(c5\), the line \(x=c\) does not satisfy the graph.Relations which recognize functionsThere is a natural means in i beg your pardon we deserve to use the relation \(x^2+y^2=25\) come construct two functions. Resolving \(x^2+y^2=25\) because that \(y\) gives\\

The graph of the first of these features is the "top half" the the circle, and also the graph that the 2nd is the "bottom half" of the circle.

comprehensive description

Example think about the ellipse \< \dfracx^216 + \dfracy^29 = 1. \>

Find two attributes whose graphs together encompass all clues on the ellipse.

Solution deal with for \(y\): \beginalign* \dfracx^216+\dfracy^29 &= 1 \\ \dfracy^29 &= 1-\dfracx^216 \\ y^2 &= \dfrac916\bigl(16-x^2\bigr) \\ y &= \pm\dfrac34\sqrt16-x^2. \endalign* Hence, \< y_1 = \dfrac34\sqrt16-x^2 \quad\textand\quad y_2 = -\dfrac34\sqrt16-x^2 \>

are two such functions.

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Note that the two graphs overlap in the points \((4,0)\) and also \((-4,0)\). This is not an issue.