Explain how we can use the idea of a systems of equations to write a piece-wise continuous function for:?

Question

the system:

y = cos x
y = e^x
y = 0.5x + 3

I understand that cos x and e^x are both 1 when x=0. So,

f(x)=cos x for x<=0
e^x for 0
But I don't understand where to go from there. I know that if I set e^x = 0.5x + 3 I have to use ln but I don't know how nore do I understand it. Can somebody please help explain what I have to do?

Answer 1

I don’t see what’s wrong about idea. Do it x=ln(0.5x+3) some 5 or 6 times starting with x=1. It converges fast! The result is x=1.29389;

Answer 2

Example of a piecewise continuous function:

f(x) = e^x if x<0
f(x) = cos x if x => 0

Here, f(x) is the same function. Its definition changes with the value of x.
It is continuous on the piece where x < 0 (x is less than 0) because e^x is everywhere continuous.
It is continuous on the piece where x => 0 (x equal to or greater than 0) because cos x is everywhere continuous.

At the "connecting" point (x=0), the limit from the left is the same as the limit from the right

e^x approaches 1 as x approaches 0
cos x approaches 1 as x approaches 0.

However it is not uniformly continuous at x = 0 (the slope jumps from 1 to 0 -- therefore, the derivative of f(x) is not continuous at that point)