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Jim Lamb ers Math 1B Fall Quarter 2004- Le ture 1 Examples
These examples orresp ond to Se tion 2.6 in the text.
Example 1 The equation y = , where is any real numb er, des rib es the onstant fun tion f (x) = , where x is any real numb er. Regardless of the input x, the output is always the same,. For instan e, y = 2 des rib es the onstant fun tion whose output is always 2. 2
Example 2 The equation y = x des rib es the identity fun tion f (x) = x, where x is any real numb er. In this ase, the output of the fun tion is always equal to the input. 2
Example 3 The equation y = x^2 des rib es the square fun tion f (x) = x^2 , where x is any real numb er. The output of the fun tion is always equal to the square of the input. Be ause the square of any real numb er is nonnegative, it follows that the range of this fun tion is the set of all nonnegative real numb ers, whi h an b e des rib ed by the inequality 0 y < 1 , or using interval notation to write [0; 1 ). On the other hand, any real numb er an b e squared, so the set of all p ossible inputs, known as the domain of the fun tion, is the set of all real numb ers. This set an b e des rib ed using the inequality 1 < x < 1 , or using interval notation, writing ( 1; 1 ). 2
Example 4 The equation y = +
p x des rib es the positive square root fun tion f (x) = +
p x, where x is a nonnegative real numb er. Sin e it is not p ossible to ompute the square ro ot of a negative numb er, the domain of f is the set of all nonnegative real numb ers, or all numb ers in the interval [0; 1 ). Sin e we are taking the p ositive square ro ot of x, the range of f is also [0; 1 ). 2
Example 5 The equation y =
p x do es not des rib e a fun tion where x is the input and y is the output, b e ause if x is nonzero, then there are two values of y that satisfy the equation: y = +
p x, and y = px. For example, if x = 4, then y ould equal 2 or 2 and still satisfy the equation. In other words, there are two outputs for every input (ex ept x = 0), whereas a fun tion has the prop erty that ea h input an only have one output. The graph of y = px is shown in Figure 1. Note that the graph do es not pass the verti al line test, b e ause it is p ossible to draw a verti al line that interse ts the graph more than on e. A urve in the xy -plane is the graph of a fun tion of x if and only if any verti al line interse ts the urve at most on e, in whi h ase we say that the graph passes the verti al line test. Note: the phrase \fun tion of x" is very imp ortant. The equation y = px does des rib e a fun tion of y , where y is the input and x is the output, b e ause for every real numb er y , there is only one x that satis es the equation, and that numb er is x = y 2. The relationship b etween x and y an b e des rib ed by the equation x = f (y ), where f (y ) = y 2 is a fun tion of y. The p oint of this remark is to emphasize that it is imp ortant to re ognize whi h variable is the input and whi h is
−2−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
−1.
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−0.
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y
y=± x1/
Figure 1: Graph of the equation y = px
the output, instead of simply assuming that x always refers to the input and y always refers to the output, as this is not always the ase. This will b e dis ussed further in a later example. 2
Example 6 The graph of the fun tion f (x) = x^2 is shown in Figure 2. This graph passes the verti al line test, b e ause for every real numb er x, there is only one real numb er y that satis es the equation y = x^2. However, if y > 0, there are two real numb ers that satisfy this equation: x = + p y and x = py. For example, if the input x is 3, then the output is 32 = 9, but if the input is 3, then the output is also ( 3)^2 = 9. It follows that a horizontal line des rib ed by the equation y = , where > 0, interse ts the graph of f (x) twi e, so we say that the graph do es not pass the horizontal line test. Su h a horizontal line is shown in Figure 2. In order for the graph of a fun tion to pass this test, it must b e the ase that any horizontal line interse ts the graph at most on e, as with the verti al line test. 2
Example 7 The graph of the fun tion f (x) = x^3 is shown in Figure 3(a). This graph passes the
−8−2 −1 0 1 2
−
−
−
0
2
4
6
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x
y
(a) y=x^3
−2 −10 −5 0 5 10
−1.
−
−0.
0
1
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y
x
(b) x=y1/
Figure 3: (a) Graph of the fun tion f (x) = x^3 , and (b) the graph of its inverse f ^1 (y ) = 3 p y
graph of f : simply inter hange the x and y o ordinates of all of the p oints on the graph of f. The graph of f ^1 is shown in Figure 3(b). By reversing the roles of x and y , the horizontal line y = 1 in Figure 3(a) app ears as a verti al line in Figure 3(b). In general, the fa t that a fun tion f passes the horizontal line test is equivalent to the fa t that its inverse fun tion f ^1 passes the verti al line test, and therefore is a fun tion. Also, the fa t that f passes the verti al line test, b e ause it is a fun tion, implies that f ^1 passes the horizontal line test, and therefore f ^1 is one-to-one, just as f is. A fun tion f and its inverse f ^1 have the e e t of undoing one another. For example, if x = 2 serves as the input to f (x) = x^3 , then the output is y = x^3 = 23 = 8. If this value of y now serves as the input to f ^1 (y ) = 3 p y , then the output is x = 3 p y = 3
p 8 = 2, and we have the numb er that we started with. In general, for any real numb er x in the domain of f , f ^1 (f (x)) = x. Similarly, for any real numb er in the range of f , whi h is also the domain of f ^1 , we have f (f ^1 (y )) = y. Note: Some prefer to always use x to refer to the input and y the output, for onsisten y, and therefore, in this example, the inverse fun tion would b e des rib ed by the equation y = f ^1 (x) = 3
p x
instead of x = 3 p y. We have not adopted this pra ti e here, b e ause it is a tually not a go o d idea to rename variables in a \real" appli ation. For example, supp ose one uses the letter m to refer to mass, and the letter v to velo ity. Certainly, these are natural, intuitive hoi es. Now, supp ose that one has an equation of the form m = f (v ) that des rib es mass as a fun tion of velo ity. If this equation is used to obtain the inverse fun tion f ^1 , then f ^1 des rib es velo ity as a fun tion of mass; that is, v = f ^1 (m). In this ase, renaming the variables m and v would only ause onfusion, rather than alleviate it. In general, it is b est to rememb er that the input of f ^1 is the output of f , and vi e versa, and not hange variable names midstream. 2
