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A scalar function is a function ƒ(x1, x2, ..., xn) that has a one-dimensional range. This is equivalent to saying that for each value of x1, x2, ..., xn, ƒ has only one value: ƒ(x1, x2, ..., xn). This property of ƒ can be written "ƒ: Rn—>R " or
"ƒ is the map/mapping from Rn to R ".

Examples of scalar functions:

• ƒ(x)=x2+3, ƒ: R—>R
• ƒ(x,y,z)=|xyz|, ƒ: R3—>R
• ƒ(x)=-x, ƒ: R—>R     Applying this function is the same as multiplying by -1

A more rigorous mathematician, instead of defining the function ƒ(x)=x2+3, would write ƒ: x—>ƒ(x) (here x2+3). This is to highlight the fact that ƒ is the function, while ƒ(x) (here x2+3) is the value of the function ƒ for a specific value of x.

A few types of scalar functions:

• Many-to-one: One or more distinct values of x correspond to a distinct value of ƒ(x).
Ex: ƒ: x—>sin(x) because for x=0, 2π, 4π, ..., ƒ(x)=0; x=π, 3π, 5π, ..., ƒ(x)=1; ...
• Two-to-one: Exactly two distinct values of x correspond to a distinct value of ƒ(x).
Ex: ƒ: x—>x2 is two-to-one except for x=0, because for x=1,-1, ƒ(x)=1; x=2,-2, ƒ(x)=4; ...
• One-to-one (aka injective): Every distinct value of x corresponds to a distinct value of ƒ(x).
Ex: ƒ: x—>2x+3, because for x=0, ƒ(x)=3; x=1, ƒ(x)=4; ...

Assuming you are dealing with real numbers, which you might not be. R is the field of real numbers, but you could just as easily use Z (the ring of integers), C (the field of complex numbers), Q (the field of rational numbers), or anything else you like.

Sources:
Bressoud, D. Second Year Calculus © 1991
http://mathworld.wolfram.com/

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