MAT 1362 โ Winter 2025
Quiz 8 โ DGD 2 (Friday) โ Solutions
Professor: Alistair Savage
Date: March 28, 2025
Question (4 points). Consider the function
f:R>0โR>0, f(x) = x+ 1
5.
(a) Is finjective? Justify your answer.
Solution: Yes, fis injective. For x1, x2โR>0, we have
f(x1) = f(x2) =โx1+ 1
5=x2+ 1
5
=โx1+ 1 = x2+ 1
=โx1=x2.
(b) Is fsurjective? Justify your answer.
Solution: No, fis not surjective. For all xโR>0, we have
x > 0 =โx+ 1 >1 =โf(x) = x+ 1
5>1
5.
Hence, for example, there is no xโR>0such that f(x) = 1
10 .
(c) Does fhave a left inverse? If it does, give one and show that it is indeed a left inverse.
Otherwise, justify why fdoes not have a left inverse.
Solution: Yes. A function has a left inverse if and only if it is injective. Since fis
injective, it has a left inverse. Define
g:R>0โR>0, g(y) = |5yโ1|.
