Functions

Excerpts

• Functions

  The essential point is that when $x$ is given, then $y$ is thereby definitely determined.

• Functions

  For the value of the function on the negative (left) side of the origin becomes endlessly great, but negative, and the function reappears on the positive (right) side as endlessly great but positive.

• Functions

  The whole difference between the older and the newer mathematics lies in the fact that vague half-metaphorical terms like “gradually” are no longer tolerated in its exact statements.

• Functions

  This is exactly the sort of definition which satisfied our mathematical forefathers and no longer satisfies modern mathematicians.

• Functions

  A man who, trusting that the mean height of the land above sea-level between London and Paris was a continuous function of the distance from London, walked at night on Shakespeare’s Cliff by Dover in contemplation of the Milky Way, would be dead before he had had time to rearrange his ideas as to the necessity of caution in scientific conclusions.

• Functions

  If a train has been travelling at the rate of twenty miles per hour, the distance ($s$ miles) gone after any number of hours, say $t$, is given by $s = 20 × t$; and $s$ is called a function of $t$.

• Functions

  With these explanations and cautions, we write $y = f(x)$, to denote that $y$ is the value of some undetermined function of the argument $x$; where $f(x)$ may stand for anything such as $x + 1$, $x^{2} - 2x + 1$, $\sin x$, $\log x$, or merely for $x$ itself.

• Functions

  Thus in $y = f(x)$, we may determine, if we choose, $f(x)$ to mean that when $x$ is an integer, $f(x)$ is zero, and when $x$ has any other value, $f(x)$ is $1$.

• Functions

  The train certainly cannot be running at forty miles per hour from 11.45 a.m. up to noon, and then suddenly, without any lapse of time, commence running at $50$ miles per hour.

• Functions

  This example brings out the fact that statements about a function $f(x)$ in the neighbourhood of a number $a$ are distinct from statements about the value of $f(x)$ when $x = a$.

Equations

• Functions

  s = 20 × t

  A train travelling at 20 miles per hour has gone s miles after t hours, so s is a function of t.

• Functions

  y = x + 1

  John's age y is one year more than Thomas's age x, so y is the function x + 1 of x.

• Functions

  y = x^{2}

  y is the square of the argument x, an example of a function of x.

• Functions

  y = 2x^{2} + 3x + 1

  y is the quadratic polynomial 2x^2 + 3x + 1 of the argument x, an example of a function.

• Functions

  y = x

  y equals the argument x itself, the simplest example of a function.

• Functions

  y = \log x

  y is the logarithm of x, quoted as an example of a function of x.

• Functions

  y = f(x)

  y is the value of some undetermined function f of the argument x; once x is given, y is definitely determined.

• Functions

  f(1) = 0

  With f defined to be 0 at integers and 1 otherwise, the value of f at the argument 1 is 0.

• Functions

  f(x) = 1

  For the example function, f(x) takes the value 1 when x is a fractional (non-integral) number.

• Functions

  f(x) = 2

  For the example function, f(x) takes the value 2 when x is an incommensurable number; this makes f discontinuous at every point.

• Functions

  p = \dfrac{1}{v}

  The value p of the function is the reciprocal of the argument v (graphed in Chapter II for positive v).

• Functions

  y = \dfrac{1}{x}

  y is the reciprocal of x for any value of x except 0; it is discontinuous at x = 0 and continuous on positive or on negative values only.

Problems

No exercises in this chapter.