04Cdomain

= Domain and Range =

Recall that


 * Domain **


 * is the set of all x values in the relation
 * x is the independent variable
 * x can be discrete or continuous


 * Range **


 * is the set of all y values in the relation
 * y is the dependent variable


 * Implied Domain **


 * When a relation is defined by a rule, the rule should include the domain.
 * If the domain is not stated, then it is implied that the domain is all the Real numbers for which the rule has meaning

For example:
 * y = x 2 + 2 for x Î ( 1, 5 ]
 * domain is 1 < x ≤  5


 * y = 2x 2 – 3x + 5
 * domain is x Î R

math . \qquad y = \sqrt{x} \\. \\ . \qquad \qquad \quad \text{domain is } x \in \big[ 0, \; \infty \big) \\ . \\ . \qquad \qquad \qquad \text{or } x \in R^+ \cup \big\{ 0 \big\} math

Interval Notation

Recall that
 * a smooth bracket indicates "less than" (<) or "greater than" (>)
 * __not__ including the end point
 * a square bracket [ ] indicates "less than or equal to" ( ≤ ) or "greater than or equal to" (__>__)
 * __including__ the end point
 * infinity ( ¥ ) always has a smooth bracket

When graphing
 * an open circle indicates not including the endpoint
 * a closed (or filled) circle indicates including the endpoint ( ≤ or ≥ )


 * circles should be drawn big enough to see


 * Note: **
 * Many textbooks and websites use __<__ or __>__ for "less than or equal to" or "greater than or equal to"

math . \qquad \text{You should always write } \leqslant \text{ or } \geqslant math


 * Examples **

Examples of intervals have been graphed on number lines

math . \qquad x > 1 \qquad or \qquad x \in \big( 1, \; \infty \big) math

math . \qquad x \leqslant 2 \qquad or \qquad x \in \big( -\infty, \; 2 \big] math

math . \qquad -1 < x \leqslant 3 \qquad or \qquad x \in \big( -1, \; 3 \big] math

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