Function Notation and Evaluation
Function Notation
Function notation replaces the generic y with a more descriptive f(x), which is read "f of x." This notation makes it clear which function you are talking about and what input you are plugging in, especially when multiple functions appear in the same problem.
Evaluating a function means substituting a specific value for x and simplifying. The process is mechanical — replace every x with the given value and compute — but it is the gateway to understanding how functions behave.
$f(x)$ ("$f$ of $x$") names the function ($f$) and shows the input ($x$). It replaces $y$: writing $f(x) = 3x - 7$ is the same as $y = 3x - 7$.
To find $f(a)$, replace every $x$ with $a$ in the formula.
$f(x) = 3x - 7$. Find $f(4)$.
$f(4) = 3(4) - 7 = 12 - 7 = 5$.
$f(x) = x^2 - 2x + 1$. Find $f(-3)$.
$f(-3) = (-3)^2 - 2(-3) + 1 = 9 + 6 + 1 = 16$.
$f(x) = 3x - 7$. Find $f(a + 1)$.
$f(a + 1) = 3(a + 1) - 7 = 3a + 3 - 7 = 3a - 4$.
Piecewise Functions
A piecewise function uses different rules on different intervals of the domain.
$g(x) = \begin{cases} x + 3 & \text{if } x < 0 \\ x^2 & \text{if } x \ge 0 \end{cases}$. Find $g(-2)$ and $g(3)$.
$g(-2) = (-2) + 3 = 1$ (since $-2 < 0$, use first rule).
$g(3) = 3^2 = 9$ (since $3 \ge 0$, use second rule).
$f(x) = 2x^2 + 1$. Find $f(0)$, $f(1)$, $f(-2)$.
$f(0) = 1$, $f(1) = 3$, $f(-2) = 2(4) + 1 = 9$.
Practice Problems
Let $f(x) = 4x - 3$, $g(x) = x^2 + 2x$, $h(x) = \begin{cases} 2x & x < 1 \\ x + 5 & x \ge 1 \end{cases}$
Show Answer Key
1. $17$
2. $-11$
3. $15$
4. $8$
5. $-3$
6. $0$
7. $4a + 5$
8. $-6$ (use $2x$ since $-3 < 1$)
9. $9$ (use $x + 5$ since $4 \ge 1$)
10. $6$ (use $x + 5$ since $1 \ge 1$)
11. $-1$
12. $4x - 3 = 13 \Rightarrow x = 4$