Question

the distance S an object travels under the influence of gravity in time t seconds is given by S(t) = 1 ÷ 2 gt + at + b where (g=9.8 acceleration due to gravity) a,b are constant check if the function S(t) is one one function (5 mark sum)​

Answer 1

The distance S an object travels under the influence of gravity in time t is given by S(t) = 1/2 gt² + at + b .

we have to check function f(t) is one -one function or not .

selects two points $t_1,t_2$ from its domain in such a way that $S(t_1)=S(t_2)$.function S(t) will be one-one function only when we will get $t_1=t_2$.

[ as you know domain of function is t ≥ 0 so, we can choose all positive real numbers ]

so, $S(t_1)=S(t_2)$

or,$\frac{1}{2}gt_1^2+at_1+b=\frac{1}{2}gt_2^2+at_2+b$

or,$\frac{1}{2}gt_1^2+at_1=\frac{1}{2}gt_2^2+at_2$

or,$\frac{1}{2}gt_1^2-\frac{1}{2}gt_2^2+at_1-at_2=0$

or,$\frac{1}{2}g(t_1^2-t_2^2)+a(t_1-t_2)=0$

or,$(t_1-t_2)\left[\frac{1}{2}g(t_1+t_2)+a\right]=0$

as you know, $\left[\frac{1}{2}g(t_1+t_2)+a\right]$ can't be negative, if a is positive constant.

so, $t_1-t_2=0$

or, $t_1=t_2$

hence, it is clear that S(t) is one-one function when constant a is positive constant.