The distance S an object travels under the influence of gravity in time t seconds is given by S(t) = 1/2.gt^2 +at +bt where, (g is the acceleration due to gravity), a, b are constants. Check if the function S (t )is one-one.
Answers
Answer:
Not one to one function
Step-by-step explanation:
For a function y=f(x) to be one to one for each and every value of x there should be an unique value of y, that is the function should be either continuously decreasing or continuously increasing for the function to be one to one.
Given:
S(t)=(gt^2)÷2 +at+bt
For the above function to be one to one, either S(t) should be continuously increasing or continuously decreasing. Differentiating the function with respect to time, we get
gt+a+b
The value could be either increasing or decreasing dependent on the value of constant a and b, so we cannot conclude that it is either continuously increasing or continuously decreasing , it may decrease for a certain values of time and suddenly increases for the remaining value of time, so each and every value of t may not have an unique solution and the function is not one to one.
Answer:
In an injective function each and every element in the domain should have a unique image/value in in the range.
In this case only t is independent value and for this equation to be injective for every value of t there should be a unique value of S.
To check we will differentiate the function and see if it is continuously decreasing or increasing.
Step 1
Differentiating the given function with respect to time.
gt+a+b
Case 1
If a and b are positive the function is continuously increasing, therefore it is an injective function.
Case 2
If a and b are negative the function is continuously decreasing till gt is less than a+b and increases after that , therefore it is not an injective function in this case.