Introduction to Functions

Transcription

1 Introduction to Functions Motivation to use function notation For the line y = 2x, when x=1, y=2; when x=2, y=4; when x=3, y=6;... We can see the relationship: the y value is always twice of the x value. There is an easier way to write this relationship. First, we give this relationship a name, say f. Next, instead of writing f ( x) = 2x. Now, instead of writing the above lines, we can write: f ( 1) = 2, f ( 2) = 4, f ( 3) = 6,... y = 2x, we write This is easier than writing "when...,..." This is one of the motivations to introduce functions. If you are new to function notation, simply treat f (x) in the same way as y. Another motivation is that we can differentiate functions. Say we have say "y", it's hard to tell which function we are referring to. With would be no confusion. y = 2x and y = 3x. When we f ( x) = 2x and g( x) = 3x, there Evaluate Functions Evaluating a function at a certain value uses the same concept as evaluating an expression, which we learned earlier. [Example 1] If f x = 2x + 3, evaluate f 1 and f 2. [Solution] Evaluating f 1 is the same as asking: When x=1, y=? In function notation, we don't use the letter y, and instead use f x. If f x = 2x + 3, then f 1 = = 5, and f 2 = = 1. This is similar to: If y = 2x + 3, then when x = 1, y = 5; and that when x = 2, y = 1. For beginners, simply treat f(x) as y.

2 The solutions f 1 = 5 and f 2 = 1 imply that the points (1,5) and (- 2,- 1) are on the graph of f x. Be careful when squares are involved. If g x = x!, then g 1 = ( 1)! = 1. If h x = x!, then h 2 = ( 2)! = 4. What relationship qualifies as a function? A relationship qualifies as a function if it gives one and only one output for each input. Look at Figure 1. Figure 1: a function and a non- function In Figure 1, there are two relationships. For f (x), we have f ( 2) = 4, f (4) = 8, f (5) = 7. This is a function, because for each input, there is one and only one output. For g (x), we have g ( 1) = 2, g(2) = 3, g(2) = 4. This is not a function because when 2 is the input, there are two different outputs. This disqualifies g (x) as a function. We can also write f (x) as a set of ordered pairs: {(2, 4), (5, 7), (4, 8)}, and g (x) as {(1, 2), (2, 3), (2, 4)}. In this situation, the symbol { } means a set of objects; the symbol ( ) means an ordered pair. Let's look at the graphs of f (x) and g (x) :

3 Figure 2: graphs of f(x) and g(x) For the graph of g (x), notice that the vertical line passes two points. This disqualifies g (x) as a function, because it has two output values for the input value x=2. We call this test Vertical Line Test. If a vertical line passes two points in the graph of a relationship, this relationship is disqualified as a function. As a comparison, the graph of f (x) passes the vertical line test, so f (x) qualifies as a function. [Example 2] Tell whether m (x) and n (x) are functions if m (x) ={(- 1, 3), (0, 3), (1, 3)}, and n (x) ={(3, - 1), (3, 0), (3, 1)}. [Solution] m (x) is a function because for each input, there is one and only one output. In other words, for input x=- 1, there is one and only one output value y=3; for input x=0, there is one and only one output value y=3; for input x=1, there is one and only one output value y=3. n (x) is not a function, because for input x=3, there are 3 different output values. We can also tell whether a relationship is a function by looking at its graph.

4 [Example 3] Tell whether h (x) and p (x) are functions based on their graphs. Figure 3: Graphs of two relationships, h(x) and p(x) [Solution] h (x) is a function because it passes vertical line test. p (x) is not a function because it fails vertical line test. Terms about Functions Assume d r = 2r models the length of a circle s diameter (value of d), given the length of the circle s radius (value of r). We say d(r) is a function of r. We call r the input value, and d(r) the output value. For example, when the input value is r = 3 cm, the output value would be d 3 = 6 cm. The variable r is also called the independent variable, and d(r) is called the dependent variable. This is because the value of d(r) depends on the value of r. Domain and Range All input values of a function make up the function's domain, and all output values of a function make up the function's range. For function m (x) = {(- 1, 3), (0, 3), (1, 3)}, the domain is {- 1, 0, 1} and the range is {3}. Note that there is no need to write duplicate values in a set. If a function's domain and range are continuous data sets, we use interval notation. See Example 4 below.

5 [Example 4] Write the domain and range of f (x) and g (x) based on their graphs. Figure 4: graphs of f(x) and g(x) [Solution] Recall that an open circle in a graph means "not including" or "exclusive", and a closed circle in a graph means "including" or "inclusive". For f (x), the domain is (2, 5], and the range is (3, 6]. For g (x), the domain is [2, 5], and the range is [3, 6]. We use set notation to write non- continuous data set, like {- 1, 0, 1}; we use interval notation to write continuous data set, like (2, 5].