Example 2 (a) Sketch the graph of p(x) = xsin x Solution For x  0 the graph of p plots the point (x,xsin x) at x times the height of the point (x,sin x) on the graph of y = sin x. In other words, the graph of p is obtained by altering the heights of the waves of height one of the graph of y = sin x to vary in height between the lines y = x and y = -x. Since p(-x)=p(x), the function p is even and its graph for x < 0 is the reflection of its graph for x>0 about the y-axis.

(b) Sketch the graph of q(x) = sin 1/x. Solution Let u=1/x. For x > 0, the values of q(x) = sin 1/x for x small are the values of sin u for u large while the values of sin 1/x for x large are the values of sin u for u small. That is, all the waves of the graph of y = sin u for u large are compressed to the right of the y-axis in the graph of q. In addition, as u approaches 0 the values of sin u approach sin 0 = 0. Hence the values of sin 1/x approach zero as x gets large, i.e. the graph of q has the x-axis as a horizontal asymptote on the right. Note that q(-x)=-q(x), and q is an odd function. Hence the graph of q for x < 0, is the reflection of the graph of q for x > 0 about the origin.

(c) Sketch the graph of r(x) = x sin 1/x. Solution The graph of r(x) = x sin 1/x for x > 0 is obtained from the graph of q(x) = sin 1/x of (b) by altering the heights of the waves, as in (a), to vary between the lines y = x and y = -x. The behavior of this graph for x large will be explained in Section 1.6. Observe that r(-x) = r(x), i.e r is an even function. Therefore the graph for x < 0 is obtained by reflecting the graph for x > 0 about the y-axis. y=x y=-x