For linear functions where the slope is zero, the function
f(x) = b is called a constant function. The textbook goes on to say that the function in question is not linear.
Why is the constant function not linear?
Definition of Constant Function
- Constant function is a linear function of the form y = b, where b is a constant.
- It is also written as f(x) = b.
- The graph of a constant function is a horizontal line.
Now, if by the strictest rule of a linear function being functions that have x as the input variable, and x is raised only to the first power, one might interpret f(x)=b as non-linear, since it is a polynomial function of degree 0. To me, this is a stretch (departure) from the difinition of the term "linear" (the graph of which is a straight line).
There are at least two different definitions of "linear" function. In basic algebra and pre-calculus, it is common to refer to any function having a straight line as graph, which can always be written y= mx+ b, as "linear". That would include m= 0, y= a constant.
In linear algebra, a linear transformation is a function satisfying f(ax+ by)= af(x)+ bf(y). For real numbers, that is of the form f(x)= mx with "constant part" 0. Of course, even in that case f(x)= 0 with both m and b equal to 0, would be "constant" linear function.
What was the title of this textbook, what course was it for, and exactly what did it say?