Function notation is a compact way to name a rule and show what input value is being used. In f(x), the letter f names the function and x represents the input. This notation matters because it lets you evaluate, compare, graph, and describe relationships clearly.
It is used throughout algebra, calculus, physics, economics, and computer science whenever one quantity depends on another.
Key Facts
- f(x) is read as "f of x" and means the output of function f when the input is x.
- If f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11.
- The input is the value placed into the function, and the output is the value produced by the rule.
- Function notation does not mean multiplication, so f(x) does not mean f times x.
- A relation is a function if each input has exactly one output.
- An equation such as y = 2x + 3 can define a function when each x-value gives one y-value, written f(x) = 2x + 3.
Vocabulary
- Function
- A function is a rule that assigns each input exactly one output.
- Input
- An input is the value placed into a function, often represented by x.
- Output
- An output is the value produced by a function after the rule is applied to the input.
- Function notation
- Function notation is a way to write a function using a name and an input, such as f(x).
- Evaluate
- To evaluate a function means to substitute a given input value and simplify to find the output.
Common Mistakes to Avoid
- Treating f(x) as multiplication is wrong because f(x) means the output of function f for input x, not f times x.
- Substituting into only part of the expression is wrong because every x in the function rule must be replaced by the given input.
- Forgetting parentheses with negative inputs is wrong because signs and exponents can change the result, such as f(-3) not being the same process as f(3).
- Assuming every equation is a function is wrong because some equations give more than one output for the same input, such as x = y^2.
Practice Questions
- 1 Let f(x) = 3x - 5. Find f(2), f(0), and f(-4).
- 2 Let g(t) = t^2 + 2t. Find g(3), g(-1), and g(a) in simplified form.
- 3 The equation x^2 + y^2 = 25 represents a circle. Explain why it does not define y as a function of x over its full graph.