Before You Start
Prerequisites
Make sure you understand what a function is (Lesson 1-1), and that you're comfortable substituting values into algebraic expressions.
Objectives
What You'll Learn
- How to evaluate f(a) for any value of a
- How to handle negative inputs, expressions, and variables as inputs
- How to evaluate functions defined by tables or graphs
Core Concept
What "Evaluate" Means
When you see f(3), it means: "plug 3 in wherever you see x." Every x gets replaced by 3, and then you simplify.
f(x) = 2x + 3
f(3) = 2(3) + 3 = 6 + 3 = 9
That's it. The notation f(3) does NOT mean "f times 3." The number in parentheses is the input, not a multiplication.
Example 1
Simple Substitution
Let f(x) = x² + 2x − 1. Find f(4).
1. f(x) = x² + 2x − 1
2. f(4) = (4)² + 2(4) − 1
3. = 16 + 8 − 1
4. = 23
Example 2
Negative Input — Be Careful!
Let g(x) = x² − 3x. Find g(−2).
1. g(x) = x² − 3x
2. g(−2) = (−2)² − 3(−2)
3. = 4 − (−6)
4. = 4 + 6 = 10
Key point: Always use brackets around your substituted value: (−2)² = 4, but −2² = −4. These are different!
Example 3
An Expression as Input
Let h(x) = 3x − 2. Find h(a + 1).
1. h(x) = 3x − 2
2. Replace every x with (a + 1):
3. h(a+1) = 3(a + 1) − 2
4. = 3a + 3 − 2
5. = 3a + 1
This is a powerful skill — evaluating at an expression, not just a number. The process is identical: replace every x and simplify.
Example 4
Evaluating From a Table
Sometimes a function is given as a table instead of a formula. Here's how to read it:
f(3) = 7 — just read across the row where x = 3.
f(0) = 1 — find x = 0, read the f(x) value.
To find x when f(x) = 13: look in the f(x) column for 13, then read the x value → x = 5.
Watch Out
Common Mistakes
- Writing f(3) = f × 3 — the parentheses mean "input," not multiplication!
- Forgetting brackets on negative numbers: f(−2) should give (−2)², not −2²
- Only replacing some x's — you must replace EVERY x in the expression
- Always use brackets around your substituted value, especially for negative numbers
- Replace ALL instances of x before you simplify
Your Turn
Try It Yourself
Q1. Given f(x) = 5x − 2, find f(6).
Show Answer
f(6) = 5(6) − 2 = 30 − 2 = 28
Q2. Given g(x) = x² − 4, find g(−3).
Show Answer
g(−3) = (−3)² − 4 = 9 − 4 = 5
Q3. Given h(x) = 2x + 1, find h(x + 3).
Show Answer
h(x+3) = 2(x+3) + 1 = 2x + 6 + 1 = 2x + 7
Key Takeaways
1-Minute Summary
- To evaluate f(a): replace every x with a, then simplify
- Always use brackets around your substituted value
- The input can be a number, a negative number, or even an expression like x + 1
- Functions can also be given as tables — just read the matching row