Question

What is a linear function?

Answer 1

A linear function is one that graphs to a straight line.

Explanation:

If a function is graphed to a straight line, it is a linear function. If there is any curve or random dip in the graph, it is not linear. None of the variables in a linear function can have an exponent or power to them. Also, a linear function must have one or two variables. If there is a third one, the third one can only be known variable or a constant. So this means the third variable can only be a lone number or variable with a known value.

An example of a linear function is y=3x. X and y are variables and 3 is the third variable with a known value.

Answer 2

A linear function is a polynomial of degree `1`.

Explanation:

A linear function of one variable has a graph which is a straight line and defined slope. Such a function can be written in slope intercept form:

`y = mx+c`

where `m` is the slope and `c` the intercept with the `y` axis.

A linear function of two real variables has a graph which is a plane in three dimensional space. The general form of such a function could be written:

`z = ax+by+c`

Since it is a function, it determines a unique value of `z` for each combination of `x` and `y`. So the plane is not parallel to the `z` axis.

More generally, a linear function of `n` real variables `x_1, x_2,..., x_n` could be written:

`y = a_1 x_1 + a_2 x_2 + ... + a_n x_n + c`

The graph of such a function is an `n` dimensional hyperplane in `n+1` dimensional space.