Basic functions

A function $f$ is a rule that assigns one output to each input. The notation $f(a)$ is not multiplication; it means substitute $a$ into the rule wherever the variable appears, then simplify. So if $f(x) = 4x^2 + 2x - 1$, then $f(-2) = 4(-2)^2 + 2(-2) - 1 = 11$.

The input can itself be an expression. To find $f(2 + 3)$ you first work out the input, $f(5)$, and then substitute.

A function can also be given as a graph. There, $f(a)$ is the $y$-coordinate of the point sitting directly above $x = a$, and the lowest or highest point of the curve gives the function's minimum or maximum.

In the figure, to read $f(3)$ go up from $x = 3$ to the curve and across to the $y$-axis: the value is $1$. The lowest point of the curve, its vertex, is the minimum. One caution: a function does not split across addition. In general $f(a + b) \neq f(a) + f(b)$ — for $f(x) = x^2$, $f(a + b) = (a + b)^2 = a^2 + 2ab + b^2$, which is not $a^2 + b^2$.

If $f(x) = 2x + 3$, what is $f(-x)$?
Substitute $-x$ for $x$ everywhere:
$f(-x) = 2(-x) + 3 = -2x + 3$.

If $g(x) = -x^2 + 2$, what is $g(2)$?
Substitute $x = 2$:
$g(2) = -(2)^2 + 2 = -4 + 2 = -2$.
On the graph this is the point on the curve above $x = 2$.

If $h(x) = 5 - 2x$, what is $h(3)$?
Substitute $x = 3$:
$h(3) = 5 - 2(3) = 5 - 6 = -1$.

If $p(x) = \frac{x + 2}{x - 1}$, what is $p(2)$?
Substitute $x = 2$:
$p(2) = \frac{2 + 2}{2 - 1} = \frac{4}{1} = 4$.

If $f(x) = 3x^2 - x + 2$, what is $f(-2)$?
Substitute $x = -2$:
$f(-2) = 3(-2)^2 - (-2) + 2 = 12 + 2 + 2 = 16$.

Reading from graphs, suppose $f(1) = 0$.
Then $g(f(1))$ means $g(0)$: read the value of $g$ where $x = 0$.
If $g(0) = 1$, then $g(f(1)) = 1$.

Does $f(2x) = 2f(x)$ for $f(x) = x^2$?
No: $f(2x) = (2x)^2 = 4x^2$,
but $2f(x) = 2x^2$.
They are not the same.