On Functions

Excerpts

• On Functions

  Such are $x^{2} + a^{2}$, $\dfrac{a + x}{a - x}$, $\log(x + y)$, $\sin 2x$.

• On Functions

  Thus if in $x^{2} + a^{2}$ $x$ only is considered as changing its value, this is called a function of $x$; if $x$ and $a$ both change, it is called a function of $x$ and $a$.

• On Functions

  Here it must be borne in mind that $\phi$ and $\psi$ do not represent numbers which multiply $x$, but are *the abbreviated directions to perform certain operations with $x$ and constant quantities*.

• On Functions

  Thus, if $\phi x = x + x^{2}$, $\phi$ is equivalent to a direction to add $x$ to its square, and the whole $\phi x$ stands for the result of this operation. Thus, in this case, $\phi(1) = 2$; $\phi(2) = 6$; $\phi a = a + a^{2}$; $\phi(x + h) = x + h + (x + h)^{2}$; $\phi \sin x = \sin x + (\sin x)^{2}$.

• On Functions

  It may be easily conceived that this notion is useless, unless there are propositions which are generally true of all functions, and which may be made the foundation of general reasoning.

• On Functions

  An expression may be a function of more quantities than one, but it is usual only to name those quantities of which it is necessary to consider a change in the value. Thus if in $x^{2} + a^{2}$ $x$ only is considered as changing its value, this is called a function of $x$; if $x$ and $a$ both change, it is called a function of $x$ and $a$.

• On Functions

  Thus in 1, the length of the radius $OB$ is a constant, the arc $AB$ is the independent variable, while $BM$, $MA$, the chord $AB$, etc., are dependent.

• On Functions

  And, as in algebra we reason on numbers by means of general symbols, each of which may afterwards be particularised as standing for any number we please, unless specially prevented by the conditions of the problem, so, in treating of functions, we use general symbols, which may, under the restrictions of the problem, stand for any function whatever.

Equations

• On Functions

  \phi x = x + x^{2}

  An illustrative function: φx is defined as x plus x squared, so φ is the direction to add x to its square.

Problems

No exercises in this chapter.