Thanks. I think I sometimes fall in the trap of trying to fully work things out 'algebraically' without thinking about them logically..
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Edit:
Some small problems here:
http://m.imgur.com/a/2eXOA
Q10ii) Is the domain in the blue wrong? Isn't the domain of gof = domain of f?
Iv) I'm having trouble finding the left half of the maximal domain?
Q8c) I'm having trouble understanding what the reverse operation is for a "reflection in the x-axis". My brain thinks the reverse of this is no reflection at all, but it gets me the wrong answer.
Thanks
For that solution (xe
x=e, or x=e/e
x), I don't think there's any way to solve it directly, just that when x =1, x=e/e
x, and as x becomes greater than 1, x>e/e
x and as x becomes less than 1, x<e/e
x so therefore x=1 is the only solution.
For your questions:
for 10 ii) I think they may be wrong. Let me know if I've made an error but:
g(f(x)) exists if the range of f is a subset of the domain of g.
Range of f = (0,infinity)
Domain of g = [0,infinity)
Therefore it exists.
The range of x values that gives the range of f to be (0,infinity) is R\{0} therefore
Domain of g(f(x)) should be R\{0}, not (0,infinity). Can anyone confirm?
iv) f o h is defined if range of h is a subset of domain of f.
Domain of f = R \ {0}
Range of g = [-1,infinity)
Therefore the g must be restricted such that the Range of g ≠ 0, ie such that range of g = [-1,0) u (0,infinity).
g(x)=0 where √2x - 1 =0
2x=1
x=1/2
So the domain is [0,infinity) without x=1/2, aka
[0,1/2) u (1/2,infinity)
8c) When you reflect in the x axis, y goes to -y, so to reverse it you divide y by -1 (or multiply). With the reverse, y goes to negative y just like when doing the initial reflection.
Hope this helps