doubt

[Rahul Sinha]

Hello,
Essential preliminary concept
Page no.69 and 70
Example 8 qii is onto or not.
I know it's not onto but was unalbe to understand what the answer in page 70 is trying to say.
Kindly help on this

[BSCG programme Mathematics]

If the function f(x) from R  to R is onto, any line y=\alpha must intersect the graph of f(x) at least at one point. 

Onto means that for any \alpha \in R, there must exist an element x_0 in R such that f(x_0)=\alpha.  This means that 

the line y=\alpha meets the graph of f(x) at the point (x_0,f(x_0)).

 Here, if we take negative values of \alpha the line y=\alpha doesn't intersect the graph  because the graph is confined to the upper part of the plane y>0. Does that make sense?

Best,

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[Rahul Sinha]

Sir
In simple language how to understand that the graph of any function is injective or surjective?
In PG 69 remark 5 it's said that any line parallel to x-axis say y=b will intersect at only one point so is injective but in PG 70 Q. E10 ii. And iii. Is said to be surjective even though it intersects at only point for y=1
Additional why Y=1 is always taken to check surjectivity in each example and solution can't we take another value instead of 1.

[BSCG programme Mathematics]

Note that, this criterion, as stated is valid only for functions from R to R.
1) A function is onto(surjective) if any line y=a meets the graph at
least at one point.
2) A function is 1-1(injective) if any line y=a meets the graph at
most at one point.
3) A function is injective and surjective if any line y=a meets the
graph at exactly one point.
I hope the above three statements will help you in clarifying the
matter. Injectivity and surjectivity are different concepts and
the criteria are also different.
1 is convenient, small value that is all. There is no special reason
here. If you try to find the roots of a polynomial equation by trial
and error, you will start
from small numbers like 1, -1, 2, 2 etc. This is similar.
Please think about why these three statements are true. This
prepackaged knowledge is of no use unless you understand
why these statements are true. Study of Mathematics is not about
learning facts; a mobile phone can learn the
facts better and faster than most humans. Study of Mathematics is
about learning how to think and reason logically. Please try to
develop
the habit.

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