Is #g(x)=10+2/x^2# a linear function and explain your reasoning?

Generally, we can think of linear functions as producing a graph that is a straight line, i.e. the slope of the line is a constant. We can see that this particular equation gives a graph of:

graph{10+(2/x^2) [-30, 30, -10, 40]} .

You can graph this type of equation in many places online, but in case you don't have access to these, here is a useful general rule.

If the equation has a exponent that is two or greater anywhere in it, it is not linear. This is also sometimes stated as having a polynomial of degree two or greater. Also, the equation when there is no variable(which is a polynomial of degree 0) is linear, but not a function.

Also, keep in mind that this rule applies to square roots, as well as cube roots and higher levels of roots. This is because we can get rid of the root by applying the inverse operation(if it is a square root you square it, as cube root you cube it,...), but in doing so you will place that power somewhere else in the equation.

So, in summary #g(x)=10 + 2/ x^2# is not a linear function because it has an exponent, and the general rule is an equation with an exponent above 1 or a root of some kind is not linear.