How to Evaluate a Function- Complete Tutorial with Examples
What Does It Mean to Evaluate a Function?
Evaluating a function means finding the output value when you know the input value. That's it. You substitute a number (or expression) into the function's formula and simplify.
Functions are written as f(x), g(x), h(x) — basically any letter with parentheses. The letter is just a name. The parentheses hold the input.
If you see f(3), it means "find the output of function f when the input is 3."
How to Evaluate a Function: Step by Step
Here's the process:
- Replace every instance of the variable with your given number
- Follow order of operations (PEMDAS)
- Simplify the expression
Let's use a simple example:
f(x) = 2x + 5
Find f(3):
Replace x with 3: 2(3) + 5
Multiply: 6 + 5
Add: 11
The answer is 11. You just evaluated the function.
Evaluating Different Types of Functions
Linear Functions
Linear functions have the form f(x) = mx + b. They're straight lines.
Example: g(x) = -4x + 7
Find g(-2):
-4(-2) + 7 = 8 + 7 = 15
Quadratic Functions
Quadratic functions have x². You square the input value.
Example: h(x) = x² - 3x + 2
Find h(4):
4² - 3(4) + 2 = 16 - 12 + 2 = 6
Polynomial Functions
These have higher powers like x³, x⁴, etc. Same process — just plug in and simplify carefully.
Example: p(x) = x³ - 2x² + x - 1
Find p(2):
2³ - 2(2)² + 2 - 1 = 8 - 8 + 2 - 1 = 1
Functions with Negative Inputs
Negative numbers need parentheses. Otherwise you'll mess up the signs.
Example: f(x) = x² + 3x
Find f(-5):
(-5)² + 3(-5) = 25 - 15 = 10
Notice the parentheses around -5. Without them, you'd get -25 - 15 = -40, which is completely wrong.
Evaluating at Expressions
Sometimes the input isn't a number — it's another expression.
Example: f(x) = 3x + 1
Find f(a + 2):
3(a + 2) + 1 = 3a + 6 + 1 = 3a + 7
The answer is a simplified expression, not a number. That's fine.
Common Mistakes That Will Cost You Points
- Forgetting parentheses with negative numbers — use them every time
- Not distributing — 2(3+4) is not the same as 2(3)+4
- Dropping negative signs — write out every step until you're confident
- Skipping the substitution step — write down f(3) = [substitution] before simplifying
Function Evaluation Quick Reference
| Function Type | General Form | Key Point |
|---|---|---|
| Linear | f(x) = mx + b | One substitution step |
| Quadratic | f(x) = ax² + bx + c | Squaring happens first |
| Cubic | f(x) = ax³ + ... | Cube the input value |
| Piecewise | Different rules for different x | Pick the correct piece |
Practice: Evaluate These Functions
Try these before checking the answers.
1. f(x) = 5x - 3. Find f(4).
2. g(x) = x² + 2x - 8. Find g(3).
3. h(x) = -2x + 9. Find h(-1).
Answers:
1. 5(4) - 3 = 20 - 3 = 17
2. 3² + 2(3) - 8 = 9 + 6 - 8 = 7
3. -2(-1) + 9 = 2 + 9 = 11
Evaluating functions is substitution and simplification. There's no magic here. Get the basics down solid and you can handle any function type.