Hi, need some help please!! Thanks!
A function f is defined by
f(x) = e^x /(1 + 2e^x)
(a) Find the domain and range of f.
(b)Find the domain and range of finverse.
(c) Sketch the graphs of f and finverse on the same axes
The domain is restricted when
$\displaystyle \frac{e^x}{1+2e^x}$
is not defined. The only time this could be undefined is if the denominator is zero. Find the domain so that the denominator is never zero.
To find the range consider what the value of f(x) is when x gets close to the limits of the domain and then consider if there is any reason why a number between the limits of the domain that would have a value of f(x) more extreme than f(x) at the limits of the domain.
The inverse of the function contains a log with a fraction in it. Do what you did with f(x) to find the domain but this time there are 2 ways a log can be not defined.
$\displaystyle log\frac{a}{b}$ is not defined when $\displaystyle {a}{b}$ is less than or equal to zero, and it is not defined when b=0
a)
x≥0, y= 1/((1/e^x)+2
y increases as x increases.
y=1/3 when x=0
y=1/2 when x=∞
if x < 0, y=1/((e^t)+2) t≥0
y decreases as t increases
y=1/3 when t=0
y=0 when t=∞ (x=-∞)
domain: all x, range: 0<y<1/2
graph: y steadily increases from 0 to ½ as x goes from -∞ to ∞.
b)
Sketch y and then look at it against a light from the other side to see (and sketch) inverse to get:
f^{-1}: domain: 0<x<1/2, range: all y
c) mirror sketch in a) about line y=x. (from calculus, you have to think about it)