Home page for accesible maths 2 Curves 2.1 Functions and their graphs 2.3 Parametrized curves

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2.2 Implicitly defined functions

We have defined the graph of the function $y=f(x)$ , $x\in I$ to be the set of points of the form $(x,f(x))$ , where $x$ ranges over the domain. Another way to write this is: $\{(x,y):x\in I,y=f(x)\}$ . Sometimes we want to define a set which looks a bit like the graph of a function, but for which $y$ can have more than one value for each $x$ .

Example 2.10 The circle with centre at the origin and radius 1 is given by the equation $x^{2}+y^{2}=1$ . In other words, it is the set of points $C=\{(x,y)\in{\mathbb{R}}^{2}:x^{2}+y^{2}=1\}$ . Note that we couldn’t write $y=f(x)$ here because for given $x\in(-1,1)$ there are two values of $y$ such that $(x,y)\in C$ : $(x,\sqrt{1-x^{2}})$ and $(x,-\sqrt{1-x^{2}})$ . This notation also helps us to avoid any concerns about the domain of the function $\sqrt{1-x^{2}}$ : if $x^{2}+y^{2}=1$ then necessarily $-1\leq x\leq 1$ .

Example 2.11 Let us look at an example we saw earlier: $y=x^{3/2}$ with $x\geq 0$ . If $y=x^{3/2}$ then $y^{2}=x^{3}$ . But if we sketch the set $\{(x,y)\in{\mathbb{R}}^{2}:y^{2}=x^{3}\}$ then we have not only $(x,x^{3/2})$ but also $(x,-x^{3/2})$ . We will see another way to describe this example later on.

Example 2.12 A similar example to a circle is an ellipse: if $a,b>0$ then

\{(x,y)\in{\mathbb{R}}^{2}:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\}

is an ellipse with centre $(0,0)$ . In fact, if $a=b$ then this is a circle with radius $a$ . We can describe an ellipse with centre $(s,t)$ in a similar way: $\{(x,y)\in{\mathbb{R}}^{2}:\frac{(x-s)^{2}}{a^{2}}+\frac{(y-t)^{2}}{b^{2}}=1\}$ .

In general, an equation of the form $F(x,y)=c$ does not give a function (at least not immediately and not always), but it describes a certain set of points in the plane. The implicit function theorem (which requires some more advanced maths, and is not covered in this module) tells us whether the equation $F(x,y)=c$ actually defines an implicit function on some neighbourhood of a given point satisfying the equation.