` ` The quickest way that you can determine if something is a linear function is by looking at the variables given.
If you do not have exponents greater than 1 for either variable, you can determine that it will be a linear function.
In this case, f(x) can be replaced merely with y.
You have y = -6x + 3. Because both of the variable terms are in the 1st degree, this will be a linear function.
Examples of non-linear functions for reference:
`y = x^2`
`y = xy`
`y = 1/x`
`y^2 = 3x `
Yes, this answer was given with assumption that the function was written in the form
y = f(x)
as opposed to as any equation for relating x and y.
Another way to determine if the function is linear is to look at the powers of x and y.
If both x and y are in the first power, then the function is linear, like the given example y= -6x + 3.
If there is y in the equation but no x (for example, y = 5), the function is still linear.
If either y or x is in the higher power of (such as `x^2` ), or there are other operations on x (such as `1/x ` or `sqrt(x)` ), the function will not be linear.
This is not completely correct. For a function to be linear, each term must have degree 1 or 0.
Consider the function xy=4; both x and y appear to the first power, but this is a hyperbola, certainly not linear. Note that xy has degree 2.
There can be several ways to do this. First, what I would do is, for all practical purposes, f(x) just means y. So, we would have:
y = -6x+3
Then, if you have learned that all linear functions can be written in the form:
y = mx+b
and see that "y = -6x + 3" is written in that form, also. Then, y = -6x+3 must be linear.
Or, if you tried to graph y = -6x+3, as in making a table:
x 1 2 3 4 5
y -3 -9 -15 -21 -27
When you plot an infinite number of points, the graph would show a straight line, thus, a linear function. For practical purposes, all you need are 2 points to draw a straight line, though. Some teachers require 3 points, the 3rd point being a check on your math for the first two.