Return to Contents
Chapter 1 - Functions
In many ways this is the most important chapter of the course - if you are able to sketch and manipulate functions with confidence, then all the other methods we will discuss will be much simpler. Ultimately, sketches in general are just diagrams designed to convey some specific bits of information whilst not worrying too much about others - function sketching is no different.
1.1 Curve Sketching
When sketching a curve, there are several key features which need to be considered.
- General Shape
- Intercepts ($x=0$ and $y=0$)
- Asymptotes
- Stationary Points ($\frac{\textrm{d}{y}}{\textrm{d}{x}}=0$)
- Inflection Points ($\frac{\textrm{d}{^2y}}{\textrm{d}{x^2}}=0$)
- Domain and Range
1.1.1 General Shape
It is very useful, before you start calculating any of the specific features, to have a picture in your mind of roughly how the curve should look. The following four plots are to help you remember some common functions that you should be familiar with. Put your finger over the colour indicators in the legend and make sure you can pair up the curves with the functions. Apologies for how busy each figure is, but this is the best way to see the patterns!
The effect of varying the index of variable:
Natural functions, illustrating that inverse functions can be constructed with simple reflections across the line $y=x$:
Trigonometric functions
Hyperbolic Functions
1.1.2 Intercepts
Once the general shape has been established, it is then often useful to be able to label certain points of interest. If you are given an explicit equation ($i.e.$ in the form $y=f(x)$), then a trivial point to find is the intercept of the vertical axis. This is evaluated by setting the independent variable to zero.
Example - For the curve $y=3x^3-47x+9$, the $y$-intercept occurs at $y=3(0)^3-47(0)+9=9$
The points at which the curve crosses the $x$-axis are called roots. For some simple equations, they can be found by inspection.
Example - The root of the curve $y=\frac{x-1}{x^2}$, can be found by considering when the function would equal zero. This will occur only when the numerator of the fraction is also zero; therefore, we need only solve $x-1=0$, giving us $x=1$.
For some other functions, we must first rearrange the equation to a form that yields the roots.
Example - The roots of the curve $y=x^2+4x-21$, can be found by first factorizing the equation to the form $y=(x-3)(x+7)$. In order to solve this equation at $y=0$, we must find the two values of $x$ that cause each of the bracketed terms to be zero. Therefore, the roots occur at $x=3$ and $x=-7$.
Furthermore, the roots of all equations of the form $y=ax^2+bx+c$ can be found using the familiar ‘quadratic formula’.
However, there remain many equations which cannot be tackled with any of the above methods. Consider, for example, the function $y=x^{3/2}-x+7$. To find the roots in this case we are forced to employ numerical methods, which are discussed in a later chapter.
1.1.3 Asymptotes
An asymptote is a straight line that is continually approached by a given curve, but does not meet it at any finite distance. Asymptotes can be vertical, horizontal or oblique (slanted), as illustrated in the following figure.
If a function can be expressed as a fraction, then a vertical asymptote will occur when the denominator equals zero. Also, if the degree of the numerator is one higher than the denominator, it may also have a slant asymptote.
Example - For the equation $y=\frac{-3x^2+2}{x-1}$, there is an easy to spot vertical asymptote when the denominator of the fraction equals zero (ie at $x=1$).
However, notice that the degree of the numerator is higher than that of the denominator, which means there may also be a slant asymptote. The next step is to perform algebraic long division, which tells us to expect an asymptote on the line $y=-3x-3$. Now that we have our slant asymptotes, we should think about whether our function will be above or below this line. Perhaps the simplest way of doing this is just to sample the pair of points either side of $x=1$ and see if they are positive or negative.
1.1.4 Stationary Points
Stationary points are where the gradient of a curve is zero. They can be found by differentiating the function and finding the values of $x$ where the differential is zero.
Example - Differentiating the function $y=x^3+x^2-8x-7$, gives the expression $\frac{dy}{dx}=3x^2+2x-8=(3x-4)(x+2)$. Stationary points occur at $(3x-4)=0$ and $(x+2)=0$, yielding $x=4/3$ and $x=-2$.
If the gradient of the function changes sign at the stationary point, then it is called a ‘turning point’. It is also possible to determine whether a turning point is a local maximum or minimum by differentiating a second time and evaluating the second differentials at each turning point. If the second differential is positive, then the point is a minimum and vice versa.
Example - Differentiating the function $y=x^3+x^2-8x-7$ twice yields $\frac{d^2y}{dx^2}=6x+2$. Taking the stationary points from the previous example, we find that evaluating the second derivative at the stationary point $x=4/3$ gives $6(4/3)+2=10$, so it is a local minimum, and similarly at the stationary point $x=-2$ gives $6(-2)+2=-10$ so it is a local maximum.
If the gradient of the function does not change sign at the stationary point, then it is called a point of ‘horizontal inflection’. Inflection points are discussed in the next section, but to visualise a curve with a stationary point that is not a turning point, think of the function $y=x^3$.
Finally, if you’d like to evaluate the $y$-coordinates of stationary points, simply substitute their $x$-coordinate back into the original equation (this might sound obvious, but people do forget!).
1.1.5 Inflection Points
An inflection point is a point on a curve at which the sign of the curvature (ie the concavity) changes. Inflection points may be stationary points (eg the function $y=x^3$), but do not have to be and they are not local maxima or local minima. They can be located by finding where the second derivative of a function equals zero.
Example - Differentiating the function $y=x^3+x^2-8x-7$ twice yields $\frac{d^2y}{dx^2}=6x+2$. Setting this to zero, we find that $6x+2=0$, which gives $x=-1/3$.
1.1.6 Domain and Range
The domain is the set of all $x$ coordinates that have a corresponding $y$ coordinate.
The range is the set of all $y$ coordinates that have a corresponding $x$ coordinate.
They can be expressed using set notation, where square brackets ‘[ ]’ signify that the point is included and round brackets ‘( )’ signify that it is excluded. By convention, infinities are consider to be excluded. If our domain has multiple regions, separated by discontinuities, then we can express this concept using the union symbol ‘$\cup$’.
Example - For the function $y=\frac{3-x}{x^2}$, as shown in the figure, there is an asymptote at $x=0$ and the global minimum (ie the lowest point) occurs at the coordinate ($6,-\frac{1}{12}$). We can therefore express the domain as $(-\infty,0)\cup(0,\infty)$ and the range as $[\frac{-1}{12},\infty)$
1.1.7 Log Axes
One application of logs that you will encounter frequently as an engineer is plotting graphs where one (‘log-linear’) or both (‘log-log’) of the axes use a log scale. For example, a log-linear plot might be required when the independent variable causes the dependant variable to range over multiple scales; whereas, the log-log plot can be used to extract ‘power law’ relationships (such as growth). The following figures plot the same three functions in each of the three graphs.
The left figure shows the functions on linear axes with which you are familiar. The middle figure, plotted with a semi-log (base 10) y-axis, makes the first two functions into straight lines. Finally, in figure on the right, by plotting the log of the functions in the appropriate base (in this case base $e$), allows the coefficient of the power to be directly measured from the graph as the gradient for the first two functions, but not for the last.
Log-log axes
In the case of log-log axis, perhaps the easiest way to see whether a function will be a straight line is to take the log of both sides of your expression and then make the following substitution.
\(X=\log(x)\qquad\&\qquad Y=\log(y)\)
And then check if this substituted function is itself linear. For example, considering the function $y=7x^2$ and then taking logs (any base is fine) of both sides, we get $\log(y)=\log(7x^2)=\log(7)+\log(x^2)=\log(7)+2\log(x)$. Now, making the above substitution, we get $Y=\log(7)+2X$. Remembering that $\log(7)$ is just a number, we see that our new expression matches the form $y=mx+c$ and so must be a straight line on a log-log scale.
1.2 Symmetry of functions
If a function is a symmetrical reflection of itself across the vertical axis, it is referred to as an `even function’. If rotating a function by 180$^\circ$ around the origin leaves it appearing unchanged, it is referred to as an ‘odd function’. If neither of the above criteria are met, then a function is neither even nor odd.
Notice that translating a curve up or down the vertical axis would not affect the evenness of an even function, but it would stop the oddness of an odd function, making it neither. Another way to thinking about this is that even functions are functions whose Taylor series would only use even powers of $x$ (zero is even!) and odd functions are functions whose Taylor series would only use odd powers of $x$. Therefore, if a function uses both even and odd powers of $x$ in its Taylor series expansion, it is neither even nor odd.
Crucially, if a function is even, then all $b_n=0$, and if a function is odd, then all $a_n=0$ (including $a_0$). Identifying this early on can save a lot of calculation time!
It turns out the functions can be decomposed into their respective even and odd constituents. Where
such that $f(x)=f_{\textrm{even}}(x)+f_{\textrm{odd}}(x)$.
Examples:
For the even function $g(x)=x^2+3$, then $g_{\textrm{even}}(x)=\frac{(x^2+3)+((-x)^2+3)}{2}=x^2+3=f(x)$ and $g_{\textrm{odd}}(x)=\frac{(x^2+3)-((-x)^2+3)}{2}=0$.
For the odd function $h(x)=x^{-1}$, then $h_{\textrm{even}}(x)=\frac{x^{-1}+(-x)^{-1}}{2}=0$ and $h_{\textrm{odd}}(x)=\frac{x^{-1}-(-x)^{-1}}{2}=x^{-1}=h(x)$.
For the neither function $p(x)=\sin(x)+3$, then $p_{\textrm{even}}(x)=\frac{(\sin(x)+3)+(\sin(-x)+3)}{2}=3$ and $p_{\textrm{odd}}(x)=\frac{(\sin(x)+3)-(\sin(-x)+3)}{2}=\sin(x)$.
What about the function $f(x)=e^x$ (consider it’s Taylor series… what’s the pattern of even and odd terms)?