Kid-friendly explanation
Function notation is like giving a machine a name and writing what you put into it. If a function is called f and you put in 3, you write f(3). The rule inside the machine tells you what comes out. For example, if f(x) = 2x, then f(3) means "double 3," so the output is 6!
Worked examples
Example 1 (basic evaluation with function notation)
Suppose a function is defined by f(x) = 3x + 5.
a) Find f(2).
To find f(2), replace x with 2 in the rule.
f(2) = 3(2) + 5 = 6 + 5 = 11.
So the input 2 has output 11.
b) If f(x) represents the total cost in dollars to rent a game for x days, and f(4) = 17, what does this mean?
It means that renting the game for 4 days costs $17.
Example 2 (combining functions and using inverses)
Let two functions be defined by g(x) = x2 + 1 and h(x) = 2x − 3.
a) Find (g + h)(x).
(g + h)(x) means add the outputs of g(x) and h(x) for the same input x.
(g + h)(x) = g(x) + h(x) = (x2 + 1) + (2x − 3) = x2 + 2x − 2.
b) Find h−1(x), the inverse of h(x), using function notation.
Start with y = h(x) = 2x − 3.
To find the inverse, swap x and y: x = 2y − 3.
Solve for y: x + 3 = 2y, so y = (x + 3) / 2.
The inverse function is h−1(x) = (x + 3) / 2.
This means h−1(x) undoes what h(x) does.
Going deeper with function notation
In high school, function notation is used with many types of functions, including linear function, quadratic, and exponential function models. You might see rules like f(x) = 4x − 1, g(x) = x2 − 5x, or P(t) = 100(1.05)t, all written using function notation.
Function notation also helps connect algebra to graphs. A point on the graph of y = f(x) is written as (x, f(x)), which shows the input x and its corresponding output. When your child evaluates f(3) or f(−1), they can find the matching point on the graph and see how the function behaves.
Later, function notation is used to represent transformations in geometry, such as mapping every point of a shape to a new location after a rotation or reflection, and to talk about inverse functions that reverse another function's rule.
Common mistakes with function notation
- Mixing up f(x) with multiplication. Some students think f(x) means f times x, so they might treat f(3) like f · 3. In function notation, f is the name of the function, and f(3) means "the output when the input is 3," not a product.
- Substituting incorrectly. Given f(x) = 2x2 − 1, a student might find f(3) as 2 · 32 − 1 = 2 · 3 − 1 = 5, forgetting to square 3 first. The correct order is 32 = 9, then 2 · 9 = 18, then 18 − 1 = 17.
- Forgetting the context. If C(t) is the temperature in degrees Celsius t minutes after starting a science experiment, a student might say C(10) is "10 degrees" instead of "the temperature after 10 minutes." Function notation always needs the meaning of both the input and the output.
How function notation shows up in everyday life
Function notation is behind many rules that feel automatic in daily life. A phone plan that charges a set amount per month, a game that gives a certain number of points for each level, or a savings account that grows by a fixed percent each year can all be written as functions using notation like C(m), P(level), or B(t). The notation is a compact way to keep track of how one quantity depends on another.