Question 1208035
.
Find the domain, range, and intercepts of the function
A(x) = 4x•sqrt{1 - x^2}. Finally, graph the function.
~~~~~~~~~~~~~~~~~~~



        In this problem,  finding domain is a simple elementary task:  the domain is   [-1,1].


        Finding  x-intercepts is also simple elementary task:  they are  -1,  0  and  1  on  x-axis.


        The key issue is to find the range.
        Tutor  Edwin made it in his post,  using derivatives and  Calculus;  it requires a painstaking technique.


        Here I will show another way to find the range, which uses simple elementary Algebra with minimum calculations.



<pre>
Let real number "t" belongs to the range.  It means that

    {{{4x*sqrt(1-x^2)}}} = t    (1)

for some value of x.  Square bot sides of (1).  You will get

    {{{16x^2*(1-x^2)}}} = t^2

    {{{16x^4 - 16x^2 + t^2}}} = 0.    (2)


You can consider equation (2) as a quadratic equation relative {{{x^2}}}.


The condition that it has a real solution for  {{{x^2}}} is this inequality for the discriminant

    {{{b^2 - 4ac}}} >= 0,  or  {{{(-16)^2 - 4*16*t^2}}} >= 0,  or  256 >= {{{4*16*t^2}}}.


It implies  {{{t^2}}} <= 4;  hence,  |t| <= 2,  or  -2 <= t <= 2.


Thus, equation (1) has a real solution if and only if  -2 <= t <= 2.


So, the range of the function  {{{4x*sqrt(1-x^2)}}}  is this interval  -2 <= t <= 2,  or t belongs  [-2,2].    <U>ANSWER</U>
</pre>

Solved.


Thus, this problem can be solved in this simple way much easier than applying Calculus.



Surely, the Calculus approach is like a heavy army tank: it is universal and works everywhere.


So, if you know Calculus and do not afraid to apply it, boldly go forward.


But if the teacher gave you similar problem long before you learn Calculus,
know and remember that this method from my post probably works.



This problem is a typical Math Olympiad level problem or a Math circle level problem 
for 9-th grades high school students, who just know Algebra, but still don't know Calculus.