Prophet Sir.........Range!

Let f: A → B.

Then the range of f, is a sub-set of B. It consists of those elements b ε B such that there exists a ε A, satisfying f(a)=b

For example f(x) = 2^x. The range of f is (0,∞).

or f(x) = x + 1/x, x ε R, x≠0. You can try and prove that the range is
R - (-2,2).

There is no one-case-fits-all way to go about finding the range. So, you could post some problems where you need help.

suppose we are asked to prove that xx+1 is invertible. then how do we do it.... without using graphs.... plz tell me....

$\boldsymbol{Ans:}$\Rightarrow$ Let $f(x)=\frac{x}{1+x}=\frac{1+x-1}{1+x}=1-\frac{1}{1+x}$\\\\ If This function is Invertable Then It must be one-to-one and onto. and Co-domain=range.\\\\

In f(x)=2^x if we subsititue x is -1/2 thn.....we get.....2^-1/2 and tht becomes 1/2^1/2 so why isits range (0 , ∞) ?

from the graph of an exponential FUNCTION..see that it never goes below the x axis or tends to meet at infinity... so range is 0 TO infinity.... you can see it logically too.

for x>=0,2^x>=1. [equality holds for x=0]
for x<0,0<2^x<1.
clearly 2^x is always >0
so range is (0,∞)