What is a linear function?

2 Answers
Nov 21, 2017

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.

Nov 21, 2017

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.