Math question pertaining to one-to one and inverse

Question

1. If a Rock falls from a height of 100 meters on Earth, the height H(in meters) after t seconds is approximately.   H(t) = 100-4.9t^2   

(a) In general, quadratic functions are not one-to-one. However, the function H s one-to-one. Why?    

(b) Find the inverse of H and verify your results.   

(c) How long will it take a rock to fall 80 meters?

 

 

 

 

 

Answer 1

Hi Nesha,

 

 1. If a Rock falls from a height of 100 meters on Earth, the height H(in meters) after t seconds is approximately. H(t) = 100-4.9t^2
(a) In general, quadratic functions are not one-to-one. However, the function H s one-to-one. Why?
(b) Find the inverse of H and verify your results.

(c) How long will it take a rock to fall 80 meters?

 

Part (a) In general, quadratic functions are not one-to-one. However, the function H s one-to-one. Why?

Quadratic equations are U-shaped and if you draw a horizontal line across the graph, the line will intersect the graph twice.  This would make the function fail what we call the "Horizontal Line"  test.

In this case, this problem is dealing with a real world situation.  Real world happens in Quadrant I.  So, it is assumed that the domain (the x-values which are t-values here) are restricted to t≥0.  With this restriction, the function H is one-to-one.

 

Part (b) Find the inverse of H and verify your results.

 

H(t) = 100-4.9t²

 

To find the inverse

1.  substitute h in the H(t)

 

h= 100-4.9t²

 

2. Switch t and h

t=100-4.9h²

 

3.  Add -100 to both sides

 

t-100=-4.9h²

 

4.  Divide both sides by -4.9

 

(t-100)=h²

-------

-4.9

 

5.  In this problem, there is a negative in the denominator which is not good form to leave.  So we multiply the entire expression by -1 to eliminate this problem.

 

(100-t)=h²

-------

4.9

 

6.  To eliminate the squared term, take the square root of both sides;

 

±√[(100-t)/4.9]=h

 

Note:  when you force a square root, you must put ± because the value could be negative or positive.  In this case, the domain of the original function was restricted to t≥0.  So the domain for the inverse is t≥0.

 

7.  Substitute inverse notation, H^-1 for h

 

H^-1(t)=√[(100-t)/4.9]     This is the inverse function.

 

 

Part (c) How long will it take a rock to fall 80 meters?

 

80=100-4.9t²

 

-20=-4.9t²

200/49=t²    take the square root of both sides

 

√(200/49)=t  simplfy

 

10√2=t

-----

   7

t=2.02  rounded to two digits.

 

 

That's it.

 

 

 

Answer 2

a) the domain of this function is all non-negative t (time) values up through that t value where the free fall terminates. This is half a parabola, so it is always decreasing, and thus 1-to-1

 

b) find the inverse by solving for t as a function of h; pay attention to the former range now being the domain (both are restricted)

 

c) H(t) = 80 = 100 - 4.9t² ...just solve it, easily.