Functions f and g are defined for x E R by
f:x --->e^x
g:x--->2x-3
(i) Solve the equation fg(x)=7
Function h is defined as gf
(ii)Express h in terms of x and state its range
(iii)Express h inverse in terms of x.
(i)x=(ln7+3)/2...its easy. take ln both sides.
(ii)h(x)=2e^x-3
its minimum value will be when the term 2e^x is minimum since 3 is constant ..2e^x can attain a minimum zero(just attain) at x=-infinity.And its maximum value will be infinity.therefore range is (-3,infinity)
(iii)for finding h inverse take log again..
Hope it helps....
Note that:
So, in this case, that means:
This leads to:
, and taking logs, we get:
Similarly,
.
Since ranges from (0,∞), it's clear that also ranges from (0,∞) and thus the range of is (-3,∞).
Consequently, is only defined on (-3,∞). Can you figure out what it is, in terms of x?