Question

How do you determine if \[f\left( x \right) = {e^{ - x}}\] is an even function or an odd function?

Answer 1

Hint: Given is an exponential function. Also we will first clarify the concept of odd and even function. Then we will apply the concept on the function so given. After that we will check which parameter the function satisfies and will declare whether it is odd or even.

Complete step by step solution:
Odd and even functions are the types of the function.
Even functions are those that are \[f\left( { - x} \right) = f\left( x \right)\].
Odd functions are those that are \[f\left( { - x} \right) = - f\left( x \right)\]
Now we will operate this on the function so given,
\[f\left( x \right) = {e^{ - x}}\]
We know that, \[{e^{ - x}} = \dfrac{1}{{{e^x}}}\]
And the exponential function of the type \[{e^x} > 0\]
So we can say that, \[\dfrac{1}{{{e^x}}}\] will be sometimes less than zero but it won't mean that it is negative unless a minus sign is observed before it.
But the exponential functions are neither positive nor negative.

Note: Here note that the square functions are generally positive but there are some cases which are odd functions. Because even function won’t mean positive value it means the same function returns for negative value of the x.