1001 Algebra Problems.PDF

Set 45 (Page 107)

705. b.The radicand of an even-indexed radical
     term (e.g., a square root) must be nonnegative
     if in the numerator of a fraction and strictly
     positive if in the denominator of a fraction.
     For the present function, this restriction takes
     the form of the inequality –x 0, which upon
     multiplication on both sides by –1, is equiva-
     lent to x 0. Hence, the domain of the func-
     tion f(x) = –xis (–∞, 0].


706. d.There is no restriction on the radicand of an
     odd-indexed radical term (e.g., a cube root) if
     it is in the numerator of a fraction, whereas
     the radicand of such a radical term must be
     nonzero if it occurs in the denominator of a
     fraction. For the present function, this restric-
     tion takes the form of the statement –1 – x≠0,
     which is equivalent to x≠–1. Hence, the
     domain of the function g(x) = is
     (–∞,–1)∪(–1,∞).


707. b.The equation y= 2 is the equation of hori-
     zontal line that crosses the y-axis at (0, 2).
     Horizontal lines have a slope of 0. This line is a
     function, since it passes the vertical line test: A
     vertical line can be drawn through the graph
     ofy= 2 at any point and will cross the graphed
     function in only one place. The domain of the
     function is infinite, but all x-values yield the
     same y-value: 2. Therefore, the range ofy= 2
     is 2.


708. b.The graph off(x) = |x| has its lowest point
     at the origin, which is both an x-intercept and
     a y-intercept. Since f(x)0 or any nonzero real
     number x, it cannot have another x-intercept.
     Moreover, a function can have only one y-
     intercept, since if it had more than one, it
     would not pass the vertical line test.
     709. b.The intersection of the graph off(x) = x^3
     and the graph of the horizontal line y= acan
     be found by solving the equation x^3 = a. Taking
     the cube root of both sides yields the solution
     x= ^3 a, which is meaningful for any real
     numbera.
     710. c.The graph off(x) = ^1 x, in fact, decreasing on
     its entire domain, not just (0,∞). Its graph is
     given here:
     711. c.The square root of a negative value is imagi-
     nary, so the value of 4x– 1 must be greater than
     or equal to 0. Symbolically, we have:

4 x– 1 0

4 x 1

x ^14 

Hence, the domain off(x) is the set of all real

numbers greater than or equal to ^14 . The small-

est value off(x) occurs at x= ^14 , and its value is

√4() – 1= ^0 = 0. So, the range of the

function is the set of all real numbers greater

than or equal to 0.

^1

4

–10 –8 –6 –4 –2 2 4 6 8 10

–2

–4

–6

–8

–10

10

8

6

4

2

^1

^3 –1 – x

ANSWERS & EXPLANATIONS–