MAT 300, Spielberg Problems G Spring 2007
1. In each of the following, decide (with proof) if what is given defines a function f:R→
R.
(i) f(x) =
√1−x2,if |x| ≤ 1
√x2−1,if |x| ≥ 1
(ii) f(x) =
√x2−4,if x≥2
√x2+ 4,if x≤2
(iii) f(x) =
1/x, if x > 0
1,if x= 0
−1/x, if x < 0
(iv) f(x) = m+n, if x=m/n, where m, n ∈Z,n6= 0, gcd(m, n) = 1
0,if x∈R\Q
(v) f(x) = mn, if x=m/n, where m, n ∈Z,n6= 0, gcd(m, n) = 1
0,if x∈R\Q
(vi) f(x) = 1/n, if x=m/n, where m, n ∈Z,n > 0, gcd(m, n) = 1
0,if x∈R\Q
(vii) f=(x2, x3) : x∈R
(viii) f=(x3, x2) : x∈R
2. In each of the following, decide (with proof) if what is given defines a function f:
[−1,1] →R.
(i) f(x) = (cos x, sin x) : x∈R
(ii) f(x) = (cos2x, sin x) : x∈R
(iii) f(x) = (cos x, sin2x) : x∈R
(iv) f(x) = (cos2x, sin2x) : x∈R
3. One-to-one and onto.
(i) Let f:R\ {1} → R\ {0}be defined by
f(x) = 1
x−1.
Prove that fis one-to-one and onto.
(ii) Let g:R\ {1} → R\ {1}be defined by
g(x) = x
x−1.
Is gone-to-one? onto? Prove your answers.
(iii) Define h: [1,∞)→Rby h(x) = √x2−1. Prove that his one-to-one. What is the
range of h? Prove your answer.
(iv) Define k:R→Rby k(x) = |x|−|x−1| − x+ 1. Is kone-to-one? onto? Prove your
answers.
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