284 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8
- (a) Prove that two functions f and g are equal (i.e., contain precisely the same
ordered pairs) if and only if dorn f = dorn g and f(x) = g(x) for all x in the
common domain.
(b) Use (a) to show that the functions f(x) = x + 4 and g(x) = (x2 - 16)/(x - 4)
are distinct.
- (a) Determine the domain and range of each of the following functions:
(i) y=x2+5
(iii) y =
(v) y = -3/x
(vii) y = (x - 2)/(2 - x)
(ix) y = (x/(l - x))'I2
(ii) y=2x2--8x+5
(iv) y = ,I=
(vi) y =^3 + cos x
(viii) y = 3/(x - 5)
(x) y=Ix-31+2
(b) Give an example of a real-valued function of a real variable for each of the
following categories:
(i) Transcendental
(ii) Algebraic, but not rational
(iii) Rational, not a polynomial. Is your example an algebraic function?
(iv) A polynomial. Is your example a rational function? an algebraic function?
- Establish whether each of the following functions is or is not one to one:
(a) f(x) = -14x + 243 (b) C(x) = 46
(c) g(x) = x4 + 5x2 (d) h(x) = x4 + 5x2, dom h = [0, co)
- (e) j(x) = x3 - x (f) k(x) = l/(x2 + 2)
(g) m(x) = x3 + x [Hint: If x, and x2 have different signs, so do m(x,) and m(x2).]
- (a) Prove that iff is a one-to-one function, then the relation f -' is a function.
(b) Prove that iff: A -+ B is a one-to-one mapping, then f -I: rng f -, A is also
a one-to-one mapping. [Note: From (a), we know that f -' is a function. You
must prove that dorn f -' = rng f, rng f -' G A, and f -' is one to one.]
- (a) Let A be any set and X E A. Prove that the inclusion mapping i,: X + A
is injective.
(b) Let f: A + B be an injective mapping and let X c A. Prove that the restric-
tion f/,: X -, B is also injective.
- In each of the following examples, compute f 0 g and g 0 f, eithet by listing
/ all the ordered pairs or by specifying domain and rule of correspondence, as
appropriate:
(a) f = f (2,413 (3, 6), (49% (5, lo)), g = ((4, 1% (6, 361, (8,641, (10, 100))
(W f = {(7, la, (5, 2), (3, ll), (8, (19 2)}, g = {(a 7), (Z5h (11,3), (8,8), (2, 1))
(a f = {(I, 11, (2,2), (3,3)}, g = {(I, 51, (3, ll), (4, 12))
f(x) = ,I-, g(x) = ex
(e) f(x) = sin x, g(x) = x3 - 5x
(f) f = ((~2)~ (2,3), (3,4), (4,5), (5% (6,7)), g = {(2,4), (3,52, (4,6), 671, (7,211
- (a) Prove formally from Definition 5 that iff and g are functions, then x E
dorn (g 0 f) if and only if x E dorn f and f(x) E dorn g.
(b) (i) Give an example of two functions f and g such that f^0 g = g^0 f.
(ii) Give an example of two functions k and h such that k 0 h # h 0 k.
