Values of the independent variable for which a function is defined

Consider the functions

$$x^2 - 2x + 5,\ \sin x,\ \arctan x$$

of the independent variable $x$. Denoting the dependent variable in each case by $y$, we may write

$$y = x^2 - 2 x + 5,\ y = \sin x,\ y = \arctan x.$$

In each case $y$ (the value of the function) is known, or, as we say, defined, for all values of $x$. This is not by any means true of all functions, as the following examples illustrating the more common exceptions will show.

$$y = \frac{a}{x - b}$$ (2.1)

Here the value of $y$ (i.e. the function) is defined for all values of $x$ except $x = b$. When $x = b$ the divisor becomes zero and the value of $y$ cannot be computed from (2.1). Any value might be assigned to the function for this value of the argument.

$$y = \sqrt{x}.$$ (2.2)

In this case the function is defined only for positive values of $x$. Negative values of $x$ give imaginary values for $y$, and these must be excluded here, where we are confining ourselves to real numbers only.

$$y = \log_a{x}. \qquad a > 0$$ (2.3)

Here $y$ is defined only for positive values of $x$. For negative values of $x$ this function does not exist (see 3.7).

$$y = \arcsin x,\ y = \arccos x.$$ (2.4)

Since sines, and cosines cannot become greater than $+1$ nor less than $-1$, it follows that the above functions are defined for all values of $x$ ranging from $-1$ to $+1$ inclusive, but for no other values.