1. First, think of a linear function such as y=f(x)=2x that hasn't any constants. The graph is a straight line that passes thru the origin(0,0) and the slope is y/x=2 (x isn't 0). The ratio of y to x is always constant(=2) means the function y is directly proportional to the variant x. CK : Substitute, e.g. 2, 4 and 2+4 for x. f(2)=2*2=4, f(4)=2*4=8. So, f(2)+f(4)=4+8=12. While f(2+4)=f(6)=2*6=12. That is, f(2)+f(4)=f(2+4) CKD. Therefore, the equation f(a)+f(b)=f(a + b) is applicable to the function y=f(x)=2x.
2. Second, think of a quadratic function such as y=f(x)=2x² that hasn't any constants in it. The graph is a parabola that is convex down and the bottom touches the origin tangentially. The slope, rate of change or tangent, is f'(x)=4x. So, the ratio of y to x changes as x varies. That is, y is not proportional to x. CK : Substitute, e.g. 2, 4 and 2+4 for x. f(2)=2*2²=8 and f(4)=2*4²=32. So, f(2)+f(4)=8+32=40. While, f(2+4)=f(6)=2*6²=72. That is, f(2)+f(4) is not equal to f(2+4). CKD. Therefor, the equation f(a)+f(b)=f(a + b) is not applicable to y=f(x)=2x².
3. In the same manner, we examine the cosine function y=f(x)=cosx. The graph is a periodic repetition of a smooth curve that fluctuates in identical waves. The amplitude is 1, the period is 2*(pi), y-intersects are {2*(pi)*n, 1) or {(2n+1)*(pi), -1} and x-intersects are {(n+½)*(pi) , 0}. (Even a 1-period of rough sketch of the graph, y=f(x)=cosx on a x-y coordinate, will help you understand this problem.) The slope, rate of change or tangent, is f'(x)=-sinx. So, the ratio of y to x varies. That means y=cosx is not proportional to x. CK : f(40)=cos40=approx. 0.766, f(30)=cos30=approx. 0.866. So, f(40)+f(30)=cos40+cos30=approx. 1.632. While f(40+30)=f(70)=cos70=approx.0.342 So that cos40+cos30 is not equal to cos70. CKD. Therefore, the answer for this question is that the equation f(a)+f(b)=f(a + b) is not applicable to cos40+cos30.
The formula for the sum of Cosine of 2 angles A and B to product form is : cosA+cosB=2cos½(A+B)*cos½(A-B) Put 40 and 30 into A and B respectively. We have : cos40+cos30=2cos35*cos5=approx. 1.632