Function Notation and Evaluation
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll strengthen your skills with function notation, evaluate functions with multiple steps, and work with functions that involve more than one operation.
The Concept
Section titled “The Concept”Function notation like f(x) tells us to take the input x and apply the rule defined by the function.
When evaluating:
- Replace every x with the given value.
- Follow the order of operations (PEMDAS/BODMAS).
- Work carefully with negative numbers and exponents.
You can also evaluate functions with expressions instead of single numbers. For example, if f(x) = 2x + 5, then f(a + 3) means replace every x with (a + 3):
Functions can represent real quantities like cost, profit, distance, or temperature.
Worked Example
Section titled “Worked Example”1. Evaluate a quadratic function
Given f(x) = 3x² − 4x + 7, find f(2).
2. Evaluate a rational function
Given g(x) = (2x + 6) / (x − 1), find g(5).
3. Evaluate with a negative input
Given h(x) = x² + 3x − 10, find h(−4).
4. Real-world evaluation
A phone plan costs 25 dollars base plus 0.15 dollars per text, so p(x) = 25 + 0.15x. Find the cost for 180 texts.
Real-World Application
Section titled “Real-World Application”Function notation is used constantly in practical situations:
- Calculating total cost: c(n) = 40 + 12n (n = number of items)
- Distance traveled: d(t) = 65t (t = hours driven at 65 mph)
- Monthly profit: p(x) = −2x² + 180x − 500 (x = units sold)
- Temperature conversion: F(C) = (9/5)C + 32
Being fluent with function notation makes it much easier to work with these real relationships as we move into more advanced topics like quadratics and exponentials.