Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/32

This page has been validated.

8

DIFFERENTIAL CALCULUS

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

	$\scriptstyle {f(x)=}$	$\scriptstyle {x^{2}-9x+14}$ ,
then	$\scriptstyle {f(y)=}$	$\scriptstyle {y^{2}-9y+14}$ .
Also	$\scriptstyle {f(a)=}$	$\scriptstyle {a^{2}-9a+14}$ ,
	$\scriptstyle {f(b+1)=}$	$\scriptstyle {(b+1)^{2}-9(b+1)+14=b^{2}-7b+6}$ ,
	$\scriptstyle {f(0)=}$	$\scriptstyle {0^{2}-9\cdot 0+14=14}$ ,
	$\scriptstyle {f(-1)=}$	$\scriptstyle {(-1)^{2}-9(-1)+14=24}$ ,
	$\scriptstyle {f(3)=}$	$\scriptstyle {3^{2}-9\cdot 3+14=-4}$ ,
	$\scriptstyle {f(7)=}$	$\scriptstyle {7^{2}-9\cdot 7+14=0}$ , etc.

Similarly, $\scriptstyle {\phi (x,y)}$ denotes a function of $\scriptstyle {x}$ and $\scriptstyle {y}$ and is read $\scriptstyle {\phi }$ of $\scriptstyle {x}$ and $\scriptstyle {y}$ .

If	$\scriptstyle {\phi (x,y)=}$	$\scriptstyle {\sin(x+y)}$ ,
then	$\scriptstyle {\phi (a,b)=}$	$\scriptstyle {\sin(a+b)}$ ,
and	$\scriptstyle {\phi ({\frac {\pi }{2}},0)=}$	$\scriptstyle {\sin {\frac {\pi }{2}}=1}$ .
Again, if	$\scriptstyle {F(x,y,z)=}$	$\scriptstyle {2x+3y-12z}$ ,
then	$\scriptstyle {F(m,-m,m)=}$	$\scriptstyle {2m-3m-12m=-13m}$ ,
and	$\scriptstyle {F(3,2,1)=}$	$\scriptstyle {2\cdot 3+3\cdot 2-12\cdot 1=0}$ .