1. FUNCTIONS
When you push a button on your phone, the tone made is actually two tones of different frequencies added together.
To obtain the graph of the sum of two or more functions, we can simply use the function property (f + g)(x) = f(x) + g(x)
So we'll find the value of each function at x and then add the values together

2. x Let's find the graph of y = x - sin x Plot these points
Finally, let’s see the graph of y = x (shown in red) and y = - sin x (shown in green) so you can see how the blue graph is the sum of these 2 functions.
Think about what the graph is trying to do. It is trying to increase linearly along the line y = x from the first function but is trying to do a negative sine wave from the second function so you see both things
Let’s see this graph zoomed out so we can see a couple of periods on either side of the origin. Look for the linear increase but also the negative sine wave since the graph is a combination of those 2 functions.

3. y = x - sin x
You can get these graphs on your calculator by putting in the combined function. IMPORTANT: Your calculator must be in RADIANS to graph trig functions. The calculator shows the function above with window -6 < x < 6 scale of  and -18 < y < 18 scale of 2.

4. x Let's find the graph of f(x) = 2 sin x + sin 2x Plot these points
The purple is 2 sin x and the red is sin 2x. See how adding the values makes the blue graph.