5.1 CONCLUSIONS INVOLVING V, BUT NOT 3 OR -, 157
it can be proved that for any real number x and positive real number a, 1x1 r a if --
and only if -a Ix <a.
(a) Use these facts to conclude that Ix + yl I 1x1 + Iyl (triangle inequality) for all
real numbers x and y. (The result was assumed and used in Example 7 of this
article.)
(6) Use the result in (a) to prove that (x - zl 2 1x1 - lzl for all real numbers x and z.
(c) Use the result in (a) to prove that lx + y +.zl < 1x1 + lyl + Izl for any three real
numbers x, y, and z. [See the Hint for Exercise 5(b).]- A real-valued function f of a real variable is said to be even if and only if
f ( - x) = f (x) for all x E R, and odd if and only if f ( - x) = - f (x) for all x E R.
(a) Prove that iff and g are even functions with domain R, then f + g, f - g,
fg, and f 0 g are even functions.
*(b) Prove that iff and g are odd functions with domain R, then f + g, f - g,
and f 0 g are odd, and fg is even.
(c) Prove that iff is odd and g is even, then f^0 g and g^0 f are both even.
lo. Let f(x) = x - (l/x).
(a) Prove that for all nonzero values of x, f (llx) = f ( -x) = - f (x).
(6) Prove that for all nonzero values of x, (f 0 f 0 f)(x) = x. - (a) Let f(x) = (ax + b)/(cx - a), where a, b, and c are arbitrary real numbers.
Show that f( f(x)) = x for all x # a/c.
(b) Suppose a, b, c, and d are real numbers satisfying the equation ad + b = bc + d.
Define functions f and g, with domain R in each case by the rules f(x) = ax + b,
g(x) = cx + d. Prove that f(g(x)) = g(f(x)) for all x E R. - A function f is said to have an absolute maximum at a point x = a if and only
if f(x) I f(a) for all x in the domain of f. Absolute minimum at a is defined
analogously.
(a) Prove, without using calculus, that f(x) = 10 - x2 has an absolute maximum
at x = 0.
*(b) Prove algebraically that g(x) = 60 + 14x - 2x2 has an absolute maximum at
x =;.
(c) Prove algebraically that if a > 0, then f(x) = ax2 + bx + c has an absolute
minimum at x = -b/2a. [Hint: In (b) and (c) write down carefully, in terms
of the preceding definition, exactly what must be proved. Use the algebraic
technique of completing the square.] - Prove or disprsve, for any three sets A, B, and C:
(a) A - (B - C) = (A - B) - C (6) (A - B)' = A' - B'
(c) Au(B-A)=AuB *(d) An(B u C)=(AnB)uC - Prove or disprove, for any real number x:
(a) sin 2x + 2 sin x = cos x + 1 (6) sin 2x cos x = sin x
(c) tan2 x + 4 = 3 sin2 x + sec2 x + 3 cos2 x - Prove or disprove, where m, n, and k are positive integers such that each of the
individual quantities is defined, that rim) = c) + (r).