[xiankai]

The functions g and h are defined by:

g(x) = In (x+1), x>-1

h(x) = 1/(x+2) + 2, x<-2

i) Explain why the composite function gh cannot be formed.
ii) Find the largest possible domain of h such that gh is a function and state the range of gh.

[Ans. (i) R_h ?D_g (where ?denotes "is not a subset of") (ii) (-?, -7/3), (-?, In3) ]

after drawing a graph, i figured out the first part of the question. but the second part stumps me.

i know for one thing, for gh to be a function, R_h =D_g (where = denotes "is a subset of")

with that in mind, i plotted a graph of h without the domain constraint, and checked the range for which y>-1.

but i end up with a range of (2, +?) and thus a domain of (-2, +?)

can anyone help me? thanks.

[Hunt]

Is this supposed to be the range of goh :? (-∞, -7/3), (-∞, In3) ?

I'm not sure I follow you, is R_h supposed to be the range of h(x) ? Or the domain of h(x) ?

For one thing, the domain of h(x) must be ] -∞, -2 [ U ] -2, +∞ [ , but if goh(x) does exist, then it has to be ] -1, +∞ [ . I don't think it's necessary to plot h(x) , it's obvious the function doesn't have any turning point and a V.A. x =-2 at ∞ and a H.A. y=2 at +∞ and - ∞. The function is strictly decreasing , but I'm not sure how this is supposed to help in determining its domain.

[xiankai]

Is this supposed to be the range of goh :? (c, (-?, In3) ?

domain, then range of gh

I'm not sure I follow you, is Rh supposed to be the range of h(x) ? Or the domain of h(x) ?

range

btw, i discovered my mistake :S i thought that the Rh was (-?, -2); when that was the domain. i've since corrected my mistake abd put it as (-?, 2) instead. now i can solve my problem ><