HSC Maths Advanced Syllabus

The Sign of a Function

Knowing the sign of a function at a certain point of a given function is important. It can allow you to determine:

A simple way to determine the sign of a function is to create a table of signs for a given equation. This is done by substituting various points, suitable for the given equation, and determining if those points are positive, negative, zero or discontinuities. A discontinuity is shown in the table of signs by text saying "undefined."

For example, the following table of signs is for the function $y=\frac{1}{x-3}$ between $-3\leq x\leq3$

$x$	-3	-1	0	1	3
Sign	-	-	-	-	undefined

The sign row, is found by substituting values of $x$ into the original equation (according to the given limits) and determining the sign of corresponding values of $y$

Vertical and Horizontal Asymptotes

Vertical asymptotes occur where a function approaches infinity or negative infinity as the input of a function approaches a certain value. They indicate points where the function is undefined. On a graph, this would be represented as a dotted line, which no curve can cross.

A simple approach to finding the vertical asymptote/s of a function is to draw up a table of signs for given function and determine where there is a discontinuity for a function. However, there is a quicker and simpler method to determine the vertical asymptotes of a function.

For a rational function written in the form $\frac{f(x)}{g(x)}$, we can find the vertical asymptotes by simply finding the solutions to $g(x)=0$. However, when doing this you must ensure the equation has been completely simplified, or you could end up finding an incorrect asymptote.

Horizontal asymptotes describe the behaviour of a function as $x$ approaches $\infty$ or $-\infty$, showing the output value that the function approaches. This is generally represented on a graph, by a dotted horizontal line, which no curve can cross. Finding the horizontal asymptotes of a function can be difficult, however, there are three simple steps to follow that can be of assistance.

For a function in the simplest form of $\frac{f(x)}{g(x)}$, the horizontal asymptote can be found when:

$$\frac{x^2-1}{x^3+2}$$ $$\frac{5x^3-1}{3x^3+2}\\\therefore\frac{5}{3}$$ $$\frac{x^3-1}{x^2+2}$$

Note that the degree of a function refers to the value of $x$ in the equation with the highest power. For example, for the function $f(x) = x^6+x^2+2$ has a degree of 6.