Question

5. Determine whether each of these functions is a bijection from R to R. (a) /()=2x-10 (b)/(a) - 4 +4 (c)/(w) - (x + 1)/(x+2) (d)/(a) 7+1

Answer 1

bijection function the fir>R and onto function (a) is one one (5): 20 -10 checking one one f(0) = 2X2 - 10 f(2=2012-10 putting for = f (62) 221 - 10 = 242-10 - 222 =222 1702 Hence, if f(-2) = f(x2) x2=X2 f is one one function. I checking onto let f(x) = such that YER 21 -10- y 21 = ng! 10 T +5 cs Scanned with CamScanner

Now, checking for y=f(x) putting value of x in fco) Ja f(x+5) = 2(45)-10 = y +10.20 = y Thus, for every JER there f(x)=y . Hence , f is onto exijt X ER such that of is it one one and is bijection. onto both so Scanned with CamScanner

(6) f@2 - 4a? +4 checking one one f(0) = 4x + 4 f(2)= 402 +4 putting of $ (27) = f(+2). 4x7 +4 = 402 +4 4a2 = 4x2 image 22 = X 0.7 57 =-72 since I does not have unique It is not one.one +(-1): 46-1214 -8 f(1) = 4(12? +4 = 8 is so this not not one one function. it is function bijection Scanned with CamScanner

(CSF.ROR : 8W) = (2322 Xt2 proving one one f(0) = Xgt 2 X2 + 2 f (02) = x2d 1 X2+2 autting f(x) = f(x2) x + 1 - X2 + 2 xit2 X 2+2 (2242) - (242) (X2d 2). (X2 + 2) - (x2+2) Xada - 1 - 2 Kaj + 2 . X2+2 1,12 = 1242 dig az one one firmetion. f is cs Scanned with CamScanner

checking onto. let f(x)= y such f(x2= 242 that YER y = x+2 x+2 x+ 2 G + 1)-(x+2) xt 2 - 3-7 = niz *+ 2 = 1 - - a = -2 1: 1-2 +27 1-9 Scanned with CamScanner

now checking y = f(x) fa)= putton x+1 x+2 a valme ?y-172 1- Y 25-1 42 Joy 24-1 +1-7 2y -] 12-23 = 12 sy it is Hence onto function, So this function is both bijection function, Scanned with CamScanner

CARA PECA) = 2° +1 checking one one one one 3 , flag = of (10) = x x + 2 + 1 putting f(x) = f(x2) o ocz + 1 = x +7 X2=X2 Hence if flag) = f(a) so X = x2 function is one one function. checking let f(x onto =y such that TGR f(x): Y x-y-1 09.) Scanned with CamScanner

y = f(x) now cheeking y = 1600 putting avalue of if() + 2 x= JJ-2 = (69-7)+7 ygt 5.7.-3+7 so f. is onto function and one one alsa and one one is onto bijtetion. L Bijection . so this function both so it is function RAR fa 22610 far= 400² +4 | +602=6+) lis +2) CS Scanner (2-83 + 2 CamScanner 465 NO YES YES. |