Overview: What Are Relations and Functions?
A relation in algebra is a set of ordered pairs. The first element of the ordered pair is the domain, and the second element in the ordered pair is the range. If every first element is paired with only one second element, or every domain has a range, it is a function. Every function is a relation, but not every relation is a function. In a science experiment, the domain is the independent variable, the one manipulated, and the range is the dependent variable, or the one measured.
Common Functions in Algebra
There are several common functions in algebra that can be represented graphically, such as the constant function, identity function, and absolute value functions. The graph for a constant function is a horizontal line with the domain (the x component) equaling all real numbers and the range (the y component) equal to the constant. Linear functions have the form y = mx +b. Therefore, for the constant function, m is equal to 0, and b equals the constant.
Inverse Relations and Functions
A relation is a set of ordered pairs. Therefore it is possible to make another set of ordered pairs from that set by switching the ordered pairs around, so that the points that were in the domain are now in the range and the points that were in the range are in the domain. The new set is called the inverse of the first set. For example, a simple set, P, has the pairs {(1,2) (3,4), (5,6), (7,8)}. The domain consists of the points [ 1,3,5,7] and the range [2,4,6,8]. Switching the domain and range creates a new set, the inverse of P, represented as P-1:{(2,1), (4,3), (6,5), and (8,7)}. Because each domain element has only one range element, the new inverse set is also a function.
Are Inverse Relations Always Functions?
Suppose we have a simple set Q. The ordered pairs that make up Q : {(1,2), (2,2), (3,2), (4,2)}. This is a function, although it is a very boring one. The domain consists of the points {1,2,3,4} and the range consists of the points (2,2,2,2). There is still one x for one y. When its inverse is created, Q-1, the set consists of {2,1),(2,2),(2,3),(2,4)}. The inverse is not a function. The domain has one element, but multiple ranges, rather than unique points.
Prediction from a Linear Model
One of the advantages to using a graphic representation of a linear function is that the model can be used to extrapolate future values of the variables in question. This also relates back to the domain as the independent variable and the range as the dependent variable. It also means that mathematical tools can be used to summarize findings from scientific experiments. For example, suppose that a gardener wants to find out what time of day is most effective for killing weeds that have grown between the bricks on the backyard patio. She devises an experiment, marking off the patio into 4 sections. Section 1 gets the treatment at 8 am, and 2 hours later, the proportion of dead weeds is measured. Section 2 gets the treatment at 9 am and 2 hours later, the proportion of dead weeds is measured, and Section 3 gets the treatment at 10 am and 2 hours later the proportion of dead weeds in that section is measured. If the relationship between the independent variable X, and the dependent variable Y follows a linear relationship, then it is possible to plot a line and predict what will be the relationship between the time of treatment at 11 am and the proportion of dead weeds in section 4.
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