A function gives exactly one output for each input. That definition came first; the notation comes now.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Naming the rule
f(x) = 2x + 1 names the rule f. f(3) means the output when the input is 3, which is 7.
A warning
f(3) is not f times 3. The brackets say "the value of f at 3".
The inputs and outputs have names.
Domain and range are the set of allowed inputs and the set of resulting outputs.
What f(x) means and does not mean
f(x) names the output of the rule f at input x. The brackets are not multiplication — f(3) does not mean f times 3. Reading it as "f of 3" or "f evaluated at 3" keeps the meaning straight.
Domain: the inputs allowed
The domain is the set of inputs for which the rule makes sense. Division by zero and square roots of negatives are the usual exclusions. Context can restrict it further — a function counting people cannot take fractional inputs.
Range: the outputs produced
The range is everything the function actually outputs as the input runs over the domain. For f(x) = x², the domain is all reals but the range is only the non-negatives. Range usually takes more thought than domain.
Why the notation earns its keep
It lets you name several rules and refer to them unambiguously, evaluate at symbolic inputs like f(a + h), and compose functions. Every one of those becomes essential in later courses, and none is possible with "y =" alone.
Step 2: Try It Yourself
Tap and try it out.
- Point(3, 7)
Step 3: Watch an Example
One step at a time.
Watch Maya Evaluate a Function
f(x) = 3x − 4. Maya finds f(5) and the domain.
- Step 1
She substitutes 5 wherever x appears: 3 × 5 − 4.
Step 4: Your Turn
Practice makes it stick.
Evaluating
Problem 1 of 2
f(x) = 4x + 3. What is f(6)?
Working Backwards
Problem 2 of 2
f(x) = 2x − 5 and f(a) = 9. What is a?
Use the Notation
1 of 4
f(x) = 5x − 2. What is f(4)?
2 of 4
f(x) = x² + 1. What is f(3)?
3 of 4
f(x) = 3x + 1 and f(a) = 16. What is a?
4 of 4
Does f(3) mean f multiplied by 3?
Step 5: Quick Check
Show what you know.
Question 1 of 2
f(x) = 6x − 7. What is f(3)?
Question 2 of 2
What is the domain of a function?
What You Learned
- f(x) names a rule; f(3) is its value at 3.
- The domain is the allowed inputs, the range is the outputs.
- f(3) is not multiplication.