Explicit function

As explained in Geometry for Programmers

An explicit equation expresses one variable directly as a function of other variables to construct points on a geometric object.

An explicit function gives one variable directly from others, providing a constructive recipe for generating points. In two dimensions, y = ax + b produces a line for chosen values of x. section 2.3.1

Use the representation with its geometric limits in mind:

• The formula y = ax + b describes nonvertical lines; a vertical line requires x = c instead. section 2.3.1
• An implicit equation can describe every line, but it does not construct points one at a time in the same direct way. section 2.3.1
• In three dimensions, producing z from x and y makes a plane, not a line. section 2.3.1
• A mathematical function assigns exactly one output to each input, distinguishing it from a general relation. section 2.4.1

Constructing a line

Choose a = 0.5 and b = 1. Evaluating y = 0.5x + 1 for successive x-values generates points on a straight line, as shown in the figure. section 2.3.1

Figure 2.15 An explicit line equation: y = 0.5x + 1

View the plotted example of an explicit line equation. section 2.3.1

How Geometry for Programmers covers it

How deep it goes

Defines direct point generation, compares implicit and canonical representations, and extends the idea from lines to planes.

Examples

Uses y = 0.5x + 1, vertical lines x = c, and a plane produced by z from x and y.

What it assumes

Builds function intuition through inputs and outputs while naming domain, codomain, and relation distinctions.

What you can do after

Evaluate and understand direct formulas for generating points on lines and recognizing when a representation describes a plane.

How it connects

Connects explicit functions with mathematical functions, implicit equations, canonical equations, lines, and planes.

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Related concepts

• line equationkind of · A line equation is an equation linear in its variables that represents a straight line in two dimensions or a linear relationship in higher dimensions.

This explainer was generated by Manning's knowledge graph from the text of Geometry for Programmers. The section links point to the parts of the book it draws on.