Function

We are given the following two equations:
1. 2.
Step 1: Add the two equations.
By adding the two
given equations, we get:
Recall that the sum of exponential terms is the definition of the hyperbolic cosine function: so we have: Since is an even function (i.e., ), it follows that is an even function.
Step 2: Subtract the two equations.
Next, subtract the second equation from the first: The difference of the exponential terms is the definition of the hyperbolic sine function: so we have: Since is an odd function (i.e., ), it follows that is an odd function.
Step 3: Conclusion
Thus, we have determined that is an even function and is an odd function.