Question

Help with functions please :D

f(x)=ax+1
g(x)=bx-1
h(x)=c(x^2)+d
where a b c d are constants and ab≠0
(for all three functions, domain of x is real number)

1. If fg=gf, show that f(x)+g(x) is independent of a and b. From here I got the relationship b=2-a (i know how to do this quesiton but i just wanna check if my relation is right)

2. From the relationship between a and b that was found in 1, find the values of a,b,c and d if hg(x)=2x^2-4x-1.

Answers: a=1, b=1, c=2, d=-3.

I tried comparing coefficient but got d=-2

Answer 1

gf = fg
b(ax + 1) - 1 = a(bx - 1) + 1
bax + b - 1 = bax - a + 1
b - 1 = 1 - a
b = 2 - a

you are correct

Answer 2

hg(x) = c((bx - 1)^2 + d)
= c(b^2x^2 - 2bx + 1 ) + d
= b^2c x^2 - 2bcx + c + d
compare 2x^2 - 4x - 1

b^2c = 2
-2bc = -1 and c + d = -1

solve these system of equations to find b,c and d

then use b = 2 - a to find a.

Answer 3

THANKS

Answer 4

yw