How do you determine if $f\left( x \right)=x+1$ is an even or an odd function?

Hint: We know that an even function is a function that satisfies the property $f\left( -x \right)=f\left( x \right).$ Also, we know that a function is said to be an odd function if and only if it satisfies the property $f\left( -x \right)=-f\left( x \right).$

Complete step-by-step solution:
Let us consider the given function $f\left( x \right)=x+1.$
We are asked to show how we determine if the function is an even or an odd function.
We know that a function is determined to be an odd function or an even function if and only if it satisfies specific properties.
A function is said to be an even function if and only if it satisfies the following property: \[f\left( -x \right)=f\left( x \right).\]
Similarly, we say that a function is an odd function if and only if it satisfies the following the property: $f\left( -x \right)=-f\left( x \right).$
Let us evaluate if the given function satisfies either of the properties.
Let us find what \[f\left( -x \right)\] which we need in both the cases.
We will get \[f\left( -x \right)=-x+1.\]
We can clearly say that this is not equal to \[f\left( x \right)=x+1.\]
We will get \[f\left( -x \right)\ne f\left( x \right).\]
We can conclude that the given function is not an even function since it does not satisfy the property\[f\left( -x \right)=f\left( x \right).\]
Similarly, we can see that $-f\left( x \right)=-\left( x+1 \right)=-x-1.$
We can see that this is also not equal to $x+1=f\left( x \right).$
We will get $f\left( -x \right)\ne -f\left( x \right).$
We can conclude that the given function is not an even function since it does not satisfy the property $f\left( -x \right)=-f\left( x \right).$
Hence, we have determined that the given function is neither even nor odd.

Note: We should always remember that there are functions that are neither an even function nor an odd function. Therefore, we should check for both even and odd functions. Here we should remember the definition of an even function and odd function.