Question

How do you determine if $f\left( x \right)=7$ is an even or odd function?

Answer 1

Hint: A function $g\left( x \right)$ is said to be an even function if it satisfies $g\left( -x \right)=g\left( x \right).$ A function $g\left( x \right)$ is said to be an odd function if it satisfies $g\left( -x \right)=-g\left( x \right).$ A function that does not satisfy the above two conditions is neither an even function nor an odd function.

Complete step by step solution:
Consider the given function $f\left( x \right)=7.$
We can see that at any point in the domain, the function value is the same. We can say this mathematically as follows:
Suppose that the function $f$ is defined on a set $\text{A}\text{.}$ Then, for any $a\in \text{A, }f\left( a \right)=7.$
So, the function value is constant. And the function is called a constant function with value $7.$
Now we are asked to find if the function is even or odd.
To find this, we can use the property that even functions satisfy and the property that odd functions satisfy.
A function $g\left( x \right)$ is said to be odd if and only if $g\left( -x \right)=-g\left( x \right).$ A function $g\left( x \right)$ is said to be even if and only if $g\left( -x \right)=g\left( x \right).$ A function that does not satisfy either of these properties is neither an odd function nor an even function.
Let us apply these properties in the given constant function.
If we are applying the necessary and sufficient condition for a function to be odd, we get $f\left( -x \right)=7\ne -7=-f\left( x \right).$
That is, the given function is not odd.
It is possible that a function can neither be odd nor be even, therefore we cannot conclude that the given function is even without applying the property.
Now we can see the given function satisfies the necessary and sufficient condition for even functions, $f\left( -x \right)=7=f\left( x \right).$
Hence, the function is even.

Note: A function is called a constant function if the function value remains the same for any point in the domain. All constant functions are even regardless of the function value. We should apply both the properties before making any conclusions for there are functions neither even nor odd.