Notation of functions

The symbol $f(x)$ is used to denote a function of $x$, and is read ``$f$ of $x$''. In order to distinguish between different functions, the prefixed letter is changed, as $F(x)$, $\phi(x)$, $f'(x)$, etc.

During any investigation the same functional symbol always indicates the same law of dependence of the function upon the variable. In the simpler cases this law takes the form of a series of analytical operations upon that variable. Hence, in such a case, the same functional symbol will indicate the same operations or series of operations, even though applied to different quantities. Thus, if

$$f(x) = x^2 - 9x + 14,$$

then

$$f(y) = y^2 - 9y + 14.$$

Also

$$f(a) = a^2 - 9a + 14,$$

$$f(b + 1) = (b + 1)^2 - 9(b + 1) + 14 = b^2 - 7b + 6,$$

$$f(0) = 0^2 - 9 \cdot 0 + 14 = 14,$$

$$f(-1) = (-1)^2 -9(-1) + 14 = 24,$$

$$f(3) = 3^2 -9 \cdot 3 + 14 = -4,$$

$$f(7) = 7^2 -9 \cdot 7 + 14 = 0,$$

etc. Similarly, $\phi(x,\ y)$ denotes a function of $x$ and $y$, and is read ``$\phi$ of $x$ and $y$''. If

$$\phi(x,\ y) = \sin(x + y),$$

then

$$\phi(a,\ b) = \sin(a + b),$$

and

$$\phi \left ( \frac{\pi}{2}, 0 \right ) = \sin \frac{\pi}{2} = 1.$$

Again, if

$$F( x,\ y,\ z ) = 2x + 3y - 12z,$$

then

$$F(m,\ -m,\ m) = 2m - 3m - 12m = - 13m.$$

and

$$F(3,\ 2,\ 1) = 2 \cdot 3 + 3 \cdot 2 - 12 \cdot 1 = 0.$$

Evidently this system of notation may be extended indefinitely.

You can define a function in Sage in several ways:

[fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\sage]

sage: x,y = var("x,y")
sage: f = log(sqrt(x))
sage: f(4)
log(4)/2
sage: f(4).simplify_log()
log(2)
sage: f = lambda x: (x^2+1)/2
sage: f(x)
(x^2 + 1)/2
sage: f(1)
1
sage: f = lambda x,y: x^2+y^2
sage: f(3,4)
25