in 3 x6, the coefficient is 3, as the coefficient takes on the sign of the operation.
Terms such as xymay not appear to have a numerical coefficient, but it is 1—a num-
ber that is not written, but assumed.
Coefficients do not have to be just numbers: In the equation 5x^3 y, the coefficient
of x^3 yis 5. But in addition, the coefficient of xis 5x^2 y, and the coefficient of yis 5x^3.
Coefficients are also seen in functions; for example, in the function f(x) 2 x, the 2 is
the coefficient.
How can functionsbe definedbased on variables?
A function having a single variable is said to be univariate;with two variables, it is
bivariate;and with more than two variables it is multivariate(although two variables
are considered multivariate by some people).
What is a linear equation?
As the term suggests, linear equations have to do with lines; and in algebra, a linear
equation means certain equations (or functions) whose graph is a line (for an exten-
sive explanation of graphs, see “Geometry and Trigonometry”). More specifically, in 139
ALGEBRA
Are there different types of functions?
Y
es, there are different types of functions—so many, in fact, that the topic of
“functions” is a book in itself. In particular, algebraic equations include
polynomial and rational expression functions. For example, polynomial equa-
tions include linear (first degree) functions, such as f(x) 2 x; a quadratic (sec-
ond degree) function example is f(x) x^2 (for more about polynomials and
degrees, see below).But “algebraic and polynomial functions” are not the only use of the term
“function”—so don’t get confused. There are also non-algebraic functions called
exponential functions, and the inverses of exponential functions, which are
called logarithmic functions. Set theory emphasizes the use of functions (for
more about functions and sets, see “Foundations of Mathematics”); and there
are trigonomic functions that include the relationships of sine, cosine, and tan-
gent functions (for more information about trigonometry, see “Geometry and
Trigonometry”). There also are continuous or discontinuous functions, tran-
scendental functions, and even real and complex functions (all this may or may
not be connected to algebra). The list goes on, but it is easy to see that mathe-
maticians love the word “function.”