What is the range of y equals 1 2 cos x?

Answer 1

Your question is fairly vague, but I'm interpreting it as:

What is the range of y=12cos(x)?

Shortform:

-12<=y<=12

[-12,12] in interval notation

Explanation:

I will explain it two ways, one way being more definite while the other is more conceptual.

To cover any misconception of the idea of "range", the mathematical definition of the range of a function is the interval covered by the output of the function. Put another way, this is the minimum and maximum "y" values that will result from plugging in the valid "x" values for the function. For some quick examples:

• The range of y=x3 is from negative infinity to infinity, which can be written in interval notation as (-infinity,infinity). This is because you can plug in any number for x and always receive a valid y. Since numbers are infinite in both the negative and positive directions and every number x corresponds to a function value y, the y-values of the function will be infinite in both directions, and therefore the range of the function is toward infinity in both directions.
• The range of y=2-x2 is from negative infinity to 2, written as (-infinity,2]. This is a parabola that opens downward, so the largest value possible for the function will be at is vertex. The vertex will occur at 0 and will have a value of 2. The parabola will continue downward forever, so the minimum value will be nonexistent since it goes toward negative infinity.
• The range of y=arctan(x) (also written as y=tan-1(x)) is from approaching -(pi)/2 to approaching (pi)/2, written as (-(pi)/2,(pi)/2). Since the inverse trigonometric functions depend on the values of the normal trigonometric functions, we can examine those functions first. Inverse tangent relies on the possible values of tangent. As you move around the unit circle, tangent increases until (pi)/2 and then decreases all the way to -(pi)/2 before beginning to increase again. You might think that the extreme tangent values are the values at (pi)/2 and -(pi)/2, but this is false because the tangent at these values is 1/0, which is undefined. But, if a real value existed for tangent at these values, then these hypothetical real values would be the extremes for the possible values of tan(x). arctan(x) returns the angles that would cause a particular tangent value, so as the input values into arctan(x) increase, they get closer and closer to (pi)/2 because this is where the largest value of tangent could possibly occur. I realize that is not very well explained, but it is an example.

Now, to tackle your function y=12cos(x).

First, you can examine this function by random substitution. you know that trigonometic functions are best examined using the unit circle (I will assume you know what this is). An easy way to examine this problem is to plug in incremental values of the unit circle for x in the function and study the resulting values of y. A table of values would look like this:

x-->y

0-->12

(pi)/6-->6sqrt(3)~10.392

(pi)/4-->6sqrt(2)~8.485

(pi)/3-->6

(pi)/2-->0

2(pi)/3-->-6

3(pi)/4-->-6sqrt(2)~-8.485

5(pi)/6-->-6sqrt(3)~-10.392

(pi)-->-12

If you continue this, you'll notice that the values keep switching back and forth from 12 to -12 then back to 12, passing through all the values in between. This is to be expected, because if you look at the graph of cosine (as well as sine), it oscillates back and forth between two values, giving it a wave-like appearance. From this you can easily surmise that the maximum value that 12cos(x) will ever reach is 12 and the minimum it will ever reach is -12, giving you the range [-12,12].

Conceptually, if you examine just the function cos(x), you realize that it oscillates back and forth between -1 and 1. So the function 12cos(x) will just take whatever results from cos(x) and multiply it by 12. Since the range of cos(x) is [-1,1], the range of 12cos(x) will just be 12 times the range of cos(x), [-12,12]. This works for any numerical amplitude modification of a sine or cosine function (putting a number in front of the function). The range of 5cos(x) would be [-5,5], the range of (pi)cos(x) would be [-(pi),(pi)], and so on for any real number.