Students evaluate functions at numbers or expressions and find the change in a function over an interval.
After completing this tutorial, you will be able to complete the following:
Recall that a function is a relation between the sets A and B such that for every , there is exactly one such that f(a)=b. We can write a function using ordered pairs (a.f(a)) where . The set A is called the domain of f and the set B is the range. Suppose that is defined by an algebraic expression (or algebraic rule) in one variable, say x. Then, we can evaluate f(x) at a real number a in its domain, by replacing each instance of x with a in the expression that defines f. For example. If is given by , then we can evaluate f(x) at 2:
We can also evaluate f(x) at a given expression a which is in the domain of f, by once again replacing every instance of x with a in the expression that defines f. For example, we can evaluate f(x) at (x+1) as follows:
Evaluating a function can give us the change in the function over an interval. In general, given and is the change in f between and . For example, suppose that we can model the speed of a plane in meters per second at time x (in seconds) by where . Now, if we want to know the change in the speed of the plane between 2 and 5 seconds, we would find the change in f on the interval from x=2 to x=5
So, the plane's speed increases 108 meters per second between 2 and 5 seconds.
We can also describe the change in a function on an interval of length k beginning at any time t. Here, we have =t and =t+k. This means that is the change in the function on an interval of length k.
These concepts can be used to interpret functional expressions that model a quantity with respect to time. Suppose P(t) models the population of a town at the end of years measured from year 2000. Let h, , and be fixed real numbers with . Then, P(t) is the town's population according to the model at the end of year 2000+t. For example, P(5) gives the town's population according to the model at the end of 2005. Also, is the population change from year to as modeled by P. For example, P(5)-P(1) gives the population change from 2001 to 2005. Finally, P(t+h) is the town's population according to the model h years after time t. So, P(t+h)-P(t) tells us the change in population from over an interval of h years (from time t to time t+h) as modeled by P. Notice that this interpretation is sensible because we took h>0.
|Approximate Time||20 Minutes|
|Pre-requisite Concepts||Students should be able to define functions, their domains, and their ranges; and determine whether a given value is in the domain of a function.|
|Type of Tutorial||Skills Application|
|Key Vocabulary||change in a function, domain, evaluate functions|