This article examines the behavior of three mathematical functions, sqrt(cos(x))*cos(300x)+sqrt(abs(x))-0.7)*(4-x*x)^0.01, sqrt(6-x^2), and -sqrt(6-x^2), from -4.5 to 4.5. We will look at the shape of each graph, the domain and range, and any interesting characteristics that appear.

## Investigating sqrt(cos(x))*cos(300x)+sqrt(abs(x))-0.7)*(4-x*x)^0.01

This function is composed of a combination of the square root of the cosine of x, the cosine of 300x, the square root of the absolute value of x, and a power of 4-x2. The graph of this function is curved, with its highest point at x=0 and its lowest point at x=-2.27 and 2.27. The domain of this function is all real numbers, and its range is from -0.7 to 0.7. This function is interesting because it has two local maxima and two local minima, which is not something that is seen often in mathematical functions.

## Examining sqrt(6-x^2) and -sqrt(6-x^2) from -4.5 to 4.5

The graphs of these two functions look very similar, with the only difference being that one is positive and the other is negative. Both functions are curved, with their highest point at x=0 and their lowest point at x=-3 and 3. The domain of both functions is from -4.5 to 4.5, and the range of both functions is from -2.45 to 2.45. These functions are interesting because they have a symmetry about the y-axis, which is not something that is seen often in mathematical functions.

In conclusion, we have examined the behavior of three mathematical functions, sqrt(cos(x))*cos(300x)+sqrt(abs(x))-0.7)*(4-x*x)^0.01, sqrt(6-x^2), and -sqrt(6-x^2), from -4.5 to 4.5. We have looked at the shape of each graph, the domain and range, and any interesting characteristics that appear. Each of these