Algebra Know-It-ALL

320 Review Questions and Answers

Question 14-8

Can we restrict the domains of relations shown by curves F or H to make either of them rep-

resent a nontrivial function of x? If so, how?

Answer 14-8

Imagine “taking slices” of curve F or curve H by considering only that portion of the

curve that falls between two movable vertical lines. This limits the values of x that apply

to the curves, thereby restricting the domain. No matter how we “slice it,” we always

get two values of y for each value of x when we do this with either F or H. The only

way we can get a function out of curve F is to bring the two vertical lines together so

they both intersect F at the extreme left point or the extreme right point of the ellipse.

That restricts the domain so severely that we get mappings of one point onto one other

point, but those are trivial functions! In the case of curve H, we can bring the two

vertical lines together so they both intersect the curve at the extreme left point. Again,

that’s a trivial result.

Question 14-9

Consider the inverses of the relations shown in Fig. 20-4. A convenient way to imagine

these inverses is to let y be the independent variable and let x be the dependent variable,

so we have mappings from values of y to values of x. Which of the inverses, defined in this

way, are functions?

–6 246

2

4

6

–2

–4

–6

–4 –2

x

y

E

F

G

H

Figure 20-4 Illustration for Questions and Answers 14-7

through 14-10.