Negative Power Functions - Dartmouth College

1 Negative Power FunctionsNegative Power Functions and Their GraphsToday we discuss Negative Power Functions . A Negative Power function is a function of the formf(x) =x n,wherenis a natural number. We could also writef(x) in the formf(x) = usual, we are going to sketch out the graphs of some sample Negative Power Functions , but before wedo that, we need to talk about the domains of Negative Power Functions . We recall that the domain of afunction is the set of possible inputs to that function. Now, for almost any real numberx, the negativepower functionx nis well-defined: just take 1 divided byxand raise it to thenth Power .

2 In the case of thenumber 0, however, the formulax nis not well-defined, because we can never divide by 0. Therefore thenumber 0 is not in the domain of any Negative Power function. We therefore say that the domain of negativepower Functions is the set of all real numbers not equal to 0. Thus Negative Power Functions are our firstexamples of Functions which do not have the entire real line as their us plot out the graphs off(x) =x 1,g(x) =x 2, andh(x) =x 3using the numerical tables below:xf(x) =x 1 4 2 1 1 2 (x) =x 2 11 (x) =x 3 4 2 1 1 8 are a few major characteristics about these graphs which require discussion.

3 First, we note that thegraphs off(x) =x 1andh(x) =x 3have point symmetry about the origin, while the graph ofg(x) =x 2has an axis of symmetry at they-axis. This suggests thatx 1andx 3are odd Functions whilex 2is aneven function, and indeed this is the case, but these Functions are like no odd or even Functions we have seenbefore because none of these Functions have values atx= , we notice the behavior of all of these Functions asxgoes to positive infinity and asxgoes tonegative infinity. In all cases, we seem to have both left and right horizontal asymptotes at thex-axis. Thisshould make sense to you: asxgets more and more positive,x 1will get closer and closer to zero, as willx 2andx 3, while remaining positive.

4 Similarly, asxgets more and more Negative ,x 1will get closer andcloser to zero, and so willx 2andx 3, withx 1andx 3always Negative andx 2always , consider the behavior of these graphs nearx= 0. Asxgets closer and closer tox= 0 from thepositive side, all three graphs become more and more vertical and all three Functions become more and morepositive. Likewise, as we approachx= 0 from the Negative side, all three graphs become more and morevertical, withx 1andx 3becoming more and more Negative , andx 2becoming more and more are our first example of vertical asymptotes, which we will discuss in more detail , note the relative behavior of the graphs ofx 1,x 2, andx 3whenxis positive.

5 Unsurprisingly,all three graphs intersect at the point (1,1), and all three have Negative slope for all positivex. When0< x <1,x 3has the highest graph, followed byx 2, followed byx 1. Whenx >1, that order is reversed:x 1is the most positive, followed byx 2, followed byx 3. Again, this should make sense to you, based onthe formulae for these three Functions , and since we have seen behavior like this before, most notably withthe positive Power of Negative Power FunctionsWe now list the major properties of Negative Power Functions which you should know:1 Domain:Iff(x) =x nfor some natural numbern, then the domain offis the set of all of the realnumbers not equal to 0.

6 We write this in set notation asDom(f) ={x R:x6= 0},where Dom(f) stands for the domain offandRis the symbol for the real numbers (this way of writingRwith double lines is used in blackboard writing by mathematicians and is called blackboard bold).You should read the symbol as in or as is a member of, and the colon means such that. Thusthe symbols inside the brackets read, the set of allxin the real numbers such thatxis not equal to0. Notice from the graphs ofx 1,x 2, andx 3that Negative Power Functions are discontinuous atx= 0:their graphs have a break in them. Also notice that, because Negative Power Functions are undefined forx= 0, they do not have derivatives atx= 0 either.

7 So Negative Power Functions are not differentiableeverywhere, although they are differentiable wherever they are defined. Evenness and Oddness:As noted above, iff(x) =x nandnis an even natural number, thenf(x)is an even function, as verified below:f( x) = ( x) n=1( x)n=1( 1)nxn=1xn=x n=f(x).Similarly, ifg(x) =x nandnis an odd natural number, theng(x) is an odd function:g( x) = ( x) n=1( x)n=1( 1)nxn=1 (xn)= (x n) = g(x).The one major caveat here is caused by the domains offandg: sincegis not defined atx= 0, we nolonger must conclude thatg(x), being an odd function, must haveg(0) = 0. The graphs ofx 1andx 3show that this is clearly not the case.

8 What is true is that ifgis defined forx, it is also definedfor x, andg( x) = g(x). In other words,gis odd where it is defined. Horizontal Asymptotes:Again, letf(x) =x nwherenis a natural number. Asxgets more andmore Negative ,f(x) gets closer and closer to 0, and asxgets more and more positive,f(x) also getscloser and closer to 0. Thus we have both a left horizontal asymptote and a right horizontal asymptote,which we write, respectively, aslimx x n= 0 andlimx x n= 0. Vertical Asymptotes:Here we have a new behavior. Letf(x) =x n, wherenis an even naturalnumber. Asxgets closer and closer (approaches) to 0 from the positive (right) side,f(x) gets moreand more positive, approaching positive infinity.

9 This is called a right vertical asymptote, and, on agraph, it is represented by the graph becoming more and more vertical as it approaches some value ofxfrom the positive side, so that eventually it looks like a vertical line, in this casex= 0. We writethe right vertical asymptote off(x) using limit notation in the following way:limx 0+f(x) = limx 0+x n= + .Likewise,f(x) also has what is called left vertical asymptote atx= 0: asxgets approaches 0 from thenegative (left) side,f(x) gets more and more positive, again approaching positive infinity. We writethe left vertical asymptote off(x) using limit notation in the following way:limx 0 f(x) = limx 0 x n= +.

10 Now letg(x) =x nwherenis an odd natural number. As we observed when we plotted the graphs ofx 1andx 3,g(x) also has both a left vertical asymptote and a right vertical asymptote atx= 0. Asxapproaches 0 from the right,g(x) becomes more and more positive, approaching positive infinity:limx 0+g(x) = limx 0+x n= + .2 Asxapproaches 0 from the right, however,g(x) becomes more and more Negative , approaching negativeinfinity:limx 0 g(x) = limx 0 x n= .So, if a function has both a left vertical asymptote and a right vertical asymptote at some value ofx,the two vertical asymptotes do not have to go in the same direction: one can go to positive infinity,and the other to Negative general, we say that a functionh(x) has a right vertical asymptote atx=aif asxapproachesafrom the positive side,h(x) either gets more and more positive, approaching positive infinity, orh(x)gets more and more Negative , approaching Negative infinity:limx a+h(x) = + orlimx a+h(x) =.