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Mathematics · Ch 15 — Functions

Representation of a Function

15.1.2

Representation of a Function

The very same function can be displayed in six different, completely equivalent ways. Using the running example 'the output exceeds twice the input by 1' as one fixed function throughout:

  1. Verbal form — a plain-English sentence describing the rule, e.g. 'output exceeds twice the input by 1'. Domain = set of inputs described; Range = set of outputs described.

  2. Arrow form on a Venn diagram — two ovals (pre-images and images) joined by arrows (Fig. 6.8). Domain = set of pre-images; Range = set of images.

  3. Ordered pair form — a set of pairs (x,y)(x,y), e.g. f={(2,5),(3,7),(4,9),(5,11)}f=\{(2,5),(3,7),(4,9),(5,11)\}. Domain = set of all first components ={2,3,4,5}=\{2,3,4,5\}; Range = set of all second components ={5,7,9,11}=\{5,7,9,11\}.

  4. Rule / formula form — an algebraic equation y=f(x)y=f(x), e.g. y=f(x)=2x+1y=f(x)=2x+1, where x∈N, 1<x<6x\in N,\ 1<x<6 (read 'f(x)f(x)' as 'f of x' or 'function of x'). Domain = the set of xx-values for which f(x)f(x) is defined; Range = the corresponding set of yy-values.

  5. Tabular form — a two-row table listing xx against yy. Domain = the xx row; Range = the yy row. …

Table 1The six representations of a function (side by side)

Verbal form: 'Output exceeds twice the input by 1'; Domain = set of inputs, Range = set of outputs

Arrow/Venn form: arrows from a set of pre-images to a set of images (see Fig. 6.8)

Ordered-pair form: f = {(2,5), (3,7), (4,9), (5,11)}; Domain = {2,3,4,5} (1st components), Range = {5,7,9,11} (2nd components)

Rule/Formula form: y = f(x) = 2x + 1, where x in N, 1 < x < 6; Domain = values of x for which f(x) is defined, Range = corresponding values of y

Tabular form: x: 2,3,4,5 | y: 5,7,9,11 …

Figure 2Fig. 6.8 — the running example as an arrow/Venn diagram

What this figure shows. A Venn-style arrow diagram for the running example function, with a left oval of pre-image (input) values and a right oval of image (output) values, arrows connecting each input to its corresponding output value of 2x+1. Labelled underneath with the reminder that the domain is the set of pre-images and the range is the set of images …

Figure 3Fig. 6.9 — the running example as a Cartesian graph

What this figure shows. A scatter of plotted points in the XY-plane corresponding to the ordered pairs of the running example function (2,5), (3,7), (4,9), (5,11), lying on a straight upward-sloping line of slope 2. Dashed guide lines show how the domain is read off as the horizontal spread (projection onto the X-axis) and the range as the vertical spread (projection onto the Y-axis) of the plo …

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