Question

The distance S an object travels under the influence of gravity in time t seconds is given by S(t) = gt +at +b 1 2 2 where, (g is the acceleration due to gravity), a, b are constants. Check if the function S t ( )is injective (1 - 1 type of function)

Answer 1

Answer with explanation:

It is given that, distance traveled (S),by an object travels under the influence of gravity in time t seconds is given by S(t)

       $=\frac{gt^2}{2}+at+b$

To ,Check whether the function,t→S(t),is Injective or not,  

If For, Different Value of t,means at different instances of time,there are different value of S, also, there may be some S,which may remain unmatched,or there are no t's (time in seconds) for that S,then the function will be Injective.

Rule for one -one

If x, and y are two different instances,for two different values of t,then

If ,S(x)=S(y)

then, x=y

Applying the rule

$=\frac{gx^2}{2}+ax+b=\frac{gy^2}{2}+ay+b\\\\\frac{g(x^2-y^2)}{2}+a(x-y)=0\\\\(x-y)\times\frac{g(x+y)}{2}+a=0\\\\ x-y=0\\\\x=y$

Hence, Proved

The function,  [tex]=\frac{gt^2}{2}+at+b[\tex] is Injective(1 - 1 type of function).