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Types of functions

The distance $$d$$ between two points $$(x,y)$$ and $$(x_0,y_0)$$ is $$$d\big((x,y),(x_0,y_0)\big) = \sqrt{(x-x_0)^2+(y-y_0)^2}.$$$

The equation of a circle centred at $$(x_0,y_0)$$ with radius $$r\gt 0$$ is $$$ (x-x_0)^2+(y-y_0)^2 = r^2.$$$ It consists of all the points that have a distance $$r$$ to the centre, since the above identity can be written as $$d\big((x,y),(x_0,x_0)\big) = r$$.

The equation of an ellipse centred at $$(x_0,y_0) $$ with radius $$a$$ and $$b$$ is $$$ \frac{(x-x_0)^2}{a^2}+\frac{(y-y_0)^2}{b^2} = 1$$$

The ellipse passes through the points $$(x_0\pm a,y_0) $$ and $$(x_0,y_0\pm b) $$.

An ellipse is the result of dilating a circle of radius 1 by a ratio of $$a$$ along the $$x$$-axis and a ratio of $$b$$ along the $$y$$-axis.

Circle (A) and ellipse (B)
Circle (A) and ellipse (B)

A parametric function expresses the coordinates of a point as a function of another variable called the parameter . This is used mainly in physics (mechanics) where the parameter is time.

In the plane, the usual notation is $$$\big(x(t),y(t)\big)$$$ for the function and $$t$$ for the parameter.

A function $$f(x)$$ is a parametric function with $$x(t)=t$$ and $$y(t) = f(t)$$.

The parametric equation for the circle is $$$ x(t) = \cos t,\quad y(t) = \sin t,\quad t\in\R.$$$ This is seen immediately as $$x^2(t)+y^2(t) = 1$$ for all $$t$$.

A rational function is the ratio of two polynomials $$$ f(x) = \frac{P(x)}{Q(x)}$$$ It can always be in reduced form, i.e. with distinct zeros for $$P$$ and $$Q$$.

The degree $$\operatorname{\textrm{deg}}(P)$$ of a polynomial $$P$$ is the highest power of $$P$$.

The degree of $$x+1$$ is $$1$$ and the degree of $$x^3-1$$ is $$3$$.

The main properties of a rational function in reduced form are as follows.

  • Domain. Everything but the zeros of $$Q$$.
  • Vertical asymptote at each zero of $$Q$$.
  • Zeros are the zeros of $$P$$.
  • Horizontal and oblique asymptotes if $$\operatorname{\textrm{deg}}(P) \le \operatorname{\textrm{deg}}(Q) + 1$$. In this case, we write $$f$$ in the form $$$ ax + b + \frac{R(x)}{Q(x)}$$$ where $$\operatorname{\textrm{deg}}(R) \le\operatorname{\textrm{deg}}(Q)-1$$. The asymptote is $$y = ax+b$$.

$$$ f(x) = \frac{x^3-x}{x^2-4}= \frac{x^3-4x+3x}{x^2-4} = x + \frac{ 3x}{x^2-4}$$$ is a rational function in reduced form, because the zeros of the numerator ($$-1$$, $$0$$ and $$1$$) and denominator ($$-2$$ and $$2$$) are distinct. The function has asymptotes $$x=-2$$, $$x=2$$ and $$y=x$$.

The equation of a hyperbola centred at $$(x_0,y_0)$$, with transverse $$x$$-axis, and radii $$a\gt0$$ and $$b\gt0$$ is $$$ \frac{(x-x_0)^2}{a^2}-\frac{(y-y_0)^2}{b^2} = 1$$$ It passes through $$(x_0\pm a,y_0) $$ and has asymptotes $$$\displaystyle \frac{y}{b} = \pm\frac{x}{a}.$$$ The two branches intersect the transverse $$x$$-axis.

We divide the equation by $$x^2$$ and send $$x\to\infty$$ to find the asymptotes $$\displaystyle \lim_{x\to\infty}\frac{y^2}{x^2} = \frac{b^2}{a^2}.$$

The equation of a hyperbola with transverse $$y$$-axis is $$$ \frac{(y-y_0)^2}{b^2} -\frac{(x-x_0)^2}{a^2}= 1.$$$

Hyperbola with transverse $$y$$-axis (A) and hyperbola with transverse $$x$$-axis (B)
Hyperbola with transverse $$y$$-axis (A) and hyperbola with transverse $$x$$-axis (B)

A curve can be defined implicitly by a relation of the form $$$ F\big(y(x),x\big) = 0.$$$

For instance, a general function $$y=f(x)$$ can be rewritten as $$y-f(x) = 0$$.

A circle is defined implicitly by the relation $$$y^2 + x^2 = 1.$$$

The curve is not defined directly as a function of $$x$$. It must be deduced from the equation, and in general corresponds to the graph of several explicit functions.

This method of defining a curve is used for those where it would be too complex to write the equation explicitly.

For instance, the circle combines the graphs of two explicit functions $$$ y_+=\sqrt{1-x^2},\quad y_-=-\sqrt{1-x^2},\qquad x\in[-1,1].$$$

Implicit and explicit equation of a circle
Implicit and explicit equation of a circle

Examples of rational functions are $$$ f(x) = \frac{ax+b}{dx+e},\quad g(x) = \frac{ax^2+bx+c}{dx+e}.$$$ We assume that $$a,d\ne 0$$ and that $$-e/d$$ is not a zero of the numerator, so that the functions are in reduced form

  • Domain. The functions are defined on $$\R\setminus \{-e/d\}$$
  • Zero. Their zeros are the zeros of the numerator.
  • Vertical asymptote: $$x=-e/d$$.
  • Horizontal asymptote for $$f$$: $$\displaystyle y = a/d$$.
  • Oblique asymptote for $$g$$. $$$\displaystyle y = \alpha x + \beta,\quad \alpha = \frac{a}{d},\; \beta=\frac{bd-ae}{d^2}.$$$

The oblique asymptote for $$g$$ is found from the computations ($$x\to\infty$$) \begin{align*} g(x) - \alpha x-\beta &= \frac{ ax^2+bx+c -\alpha d x^2 - (\alpha e + \beta d)x - \beta e}{ dx+e }\\ &= \frac{cd^2-bde+ae^2}{d^2(dx+e)}\to 0. \end{align*}

Rational functions: $$f(x)=\frac{x}{x+1}$$ and $$g(x)=\frac{x^2-1}{x}$$
Rational functions: $$f(x)=\frac{x}{x+1}$$ and $$g(x)=\frac{x^2-1}{x}$$