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Graphical representation of the domain and range of real functions

A function specifies a rule by which an input is converted to a unique output.

More precisely:

A function $f$ is a rule that assigns to each element $x$ in a set $D$ exactly one element, called $f(x)$, in a set $E$.

The domain of a function is the set of all possible $x$ values that can be used as inputs, and the range is the set of all possible $f(x)$ values that arise as outputs.

It's helpful to think of a function as a machine (see Figure 1). If $x$ is in the domain of the function then when enters the machine, it is accepted as an input and the machine produces an output according to the rule of the function. Thus we can think of the domain as the set of all possible inputs $x$ and the range as the set of all possible outputs  $f(x)$.

The most common method for visualising a function is its graph; which consists of all points $(x,y)$ in the coordinate plane such  that  $y=f(x)$ and $x$ is in the domain of $f$.

The graph of a function $f$ gives us a useful picture of the behavi…

A geometrical approach to the concepts of speed and acceleration (Part I)

1. Introduction

1.1 The concepts of speed and acceleration
How good is your understanding of speed and acceleration? Would it stand up under cross-examination in court? Here are some questions that you might be asked by a court prosecutor to test your credibility as an expert witness:
1.Speed is understood as distance divided by time. What distance is being referred to here? What time is being referred to here?
2.Is this what your car’s speedometer is measuring when it registers, for example, 60 km/hr? To what distance and time would it be referring?
3.Until the introduction of digital technology in the 1980s, cars used eddy current speedometers (you can google this). Did these measure any actual distances and times? What did they measure?
4.How would you measure the change in speed of a vehicle?
5.What do you think acceleration means? (‘Getting faster’ isn’t an expert answer and would reflect badly on your credibility. You would be expected to give a numerical definition.)
1. 2 Galileo…