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?
If your textbook says that the constant function f(x)=b is not linear, it shouldn't have left it at that. Otherwise, I would question the integrity of the whole book. I would define the constant function this way:
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?