Question

Determine whether the function \(f\left( t \right)\) which reflects the height of the football at \(t\) seconds after kickoff is one to one or not.

Short answer

The function is not one-to-one.

Step-by-step solution

Given data

The given function is height of the football at \(t\) seconds after kickoff.

Concept of functions

The simplest definition is an equation will be a function if, for any \({\rm{x}}\) in the domain of the equation (the domain is all the \({\rm{x}}\)'s that can be plugged into the equation), the equation will yield exactly one value of \({\rm{y}}\) when we evaluate the equation at a specific \({\rm{X}}\).

Simplify the expression

It is known that the football will reach ground in few seconds after kickoff.

And hence the football will reach the same height at two different times.
Thus, the graph of the function \(f(t) = \) Height of the football after kickoff is shown below in Figure.

From Figure, it is observed that the horizontal line intersects the curve at two points, which means it fails the horizontal line test.

Therefore, it is concluded that the function\(f(t) = \) Height of the football after kickoff is not a one-to-one function.