Mathematics · Ch 15 — Functions

Constant Function and Identity Function

15.1.5.1

Constant Function and Identity Function

Some basic functions (all with $f:R\to R$ unless stated otherwise).

1. Constant function. Form: $f(x)=k$ , for a fixed $k\in R$ . Every input, without exception, produces the same output $k$ . Example: $f(x)=2$ (Fig. 6.17). Domain: $R$ (or $(-\infty,\infty)$ ); Range: $\{2\}$ — just the single value $k$ . …

Figure 1Fig. 6.17 — graph of the constant function f(x)=2

What this figure shows. A horizontal straight line drawn at height y=2 across the entire XY-plane, extending indefinitely left and right, with no dependence on x at all. Illustrates that a constant function's graph is always a flat horizontal line at the height of the constant, with domain R and range the single value {2} …

Figure 2Fig. 6.18 — graph of the identity function f(x)=x

What this figure shows. A straight line through the origin rising at a 45-degree angle (slope 1), passing through points such as (-3,-3), (-1,-1), (0,0), (1,1), (2,2), extending indefinitely in both directions. Illustrates that the identity function's graph is the line y=x itself, with domain R and range R. …

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