Functions and Their Graphs

If the domain consists of a collection of real numbers (perhaps all real numbers) and the range is a collection of real numbers, then the function is called a numerical function. The rule for a numerical function can be given in several ways, but it is usually written as a formula. If the rule for a numerical function, $f$, is "the output is the input number multiplied by itself", then we could write the rule as $f(x) = x.x = x^2$ . The use of an "$x$" to represent the input is simply a matter of convenience and custom. We could also represent the same function by $f(a) = a^2$, $f (€ \# ) = \#^2$ or $f( input ) = ( input )^2$ .

For the function f defined by $f(x) = x^2 – x$ , we have that $f(3) = 3^2 – 3 = 6$, $f(.5) = (.5)^2 – (.5) = –.25$, and $f(–2) = (–2)^2 – (–2) = 6$. Notice that the two different inputs, 3 and –2, both lead to the output of 6. That is allowable for a function. We can also evaluate f if the input contains variables. If we replace the "$x$" with something else in the notation "$f(x)$", then we must replace the "$x$" with the same thing everywhere in the equation:
$f(c) = c^2 – c , f(a+1) = (a+1)^2 – (a+1) = (a^2 + 2a + 1) – (a + 1) = a^2 + a$,
$f(x+h) = (x+h)^2 – (x+h) = (x^2+2xh+h^2) – (x+h)$ , and, in general, $f(input) = (input)^2 – (input)$.

For more complicated expressions, we can just proceed step–by–step:

$\frac{f(x+h) – f(x)}{h} = \dfrac{{(x+h)^2 – (x+h)} – {x^2 – x}}{h} = \dfrac{{(x^2+2xh+h^2) – (x+h)} – {x^2 – x}}{h}$

$= \dfrac{2xh + h^2 – h}{h} = \dfrac{h(2x + h – 1)}{h} = 2x + h – 1$.

Practice 2: For the function $g$ defined by $g(t) = t^2 – 5t$ , evaluate $g(1), g(–2), g(w+3), g(x+h), g(x+h) – g(x)$, and $\frac{g(x+h) – g(x)}{h}$.