Evaluating Functions Practice- Exercises
What Evaluating Functions Actually Means
You have a function. You have a number. You plug the number into the function and get an answer. That's it. That's evaluating functions.
No philosophical revelations here. No "unlocking the power of mathematical thinking." Just substitution and arithmetic done correctly.
Function Notation: The Bare Minimum You Need
Before you can evaluate anything, you need to understand what you're looking at.
f(x) means "f of x" — it's not f times x. This trips up more students than it should.
The x inside the parentheses is your input. Whatever number (or expression) goes there gets processed by the function's rule.
f(3) means: take the number 3, feed it into function f, and write down what comes out.
The Three Parts of Any Function
- Input — what you start with (usually x)
- Rule — what the function does to the input
- Output — the result (also called f(x) or y)
How to Evaluate Functions: Step by Step
Here's the actual process. No magic, just method.
Step 1: Identify the Input Value
If you see f(5), your input is 5. If you see f(a + 2), your input is the entire expression a + 2.
Step 2: Substitute Everywhere
Replace every instance of the variable with your input. Every single one.
Example: If f(x) = 3x + 7 and you need f(4):
Replace x with 4 → f(4) = 3(4) + 7
Step 3: Simplify
Do the arithmetic. 3(4) = 12. 12 + 7 = 19.
f(4) = 19.
Practice Exercises: Evaluating Functions
Work through these. Check your answers only after you've attempted each one.
Exercise Set 1: Basic Evaluation
1. Given f(x) = 2x + 5, find f(3).
2. Given g(x) = x² - 4, find g(2).
3. Given h(x) = 8/x, find h(4).
4. Given p(x) = 7, find p(1000).
Exercise Set 2: Negative Numbers and Fractions
5. Given f(x) = x² + 2x - 1, find f(-3).
6. Given m(x) = (x + 1)/(x - 1), find m(3).
7. Given k(x) = √x, find k(16).
Exercise Set 3: Expressions as Inputs
8. Given f(x) = 2x + 1, find f(a).
9. Given f(x) = x² - 3, find f(2h).
10. Given f(x) = 5x + 2, find f(x + 3).
Answers
1. f(3) = 2(3) + 5 = 6 + 5 = 11
2. g(2) = 2² - 4 = 4 - 4 = 0
3. h(4) = 8/4 = 2
4. p(1000) = 7 (constant functions return the same value regardless of input)
5. f(-3) = (-3)² + 2(-3) - 1 = 9 - 6 - 1 = 2
6. m(3) = (3 + 1)/(3 - 1) = 4/2 = 2
7. k(16) = √16 = 4
8. f(a) = 2a + 1
9. f(2h) = (2h)² - 3 = 4h² - 3
10. f(x + 3) = 5(x + 3) + 2 = 5x + 15 + 2 = 5x + 17
Common Mistakes That Cost You Points
- Forgetting parentheses: f(-3) ≠ -3². It's (-3)² = 9. The parentheses matter.
- Skipping substitution steps: Trying to do it mentally and missing terms.
- Order of operations: Exponents before multiplication, multiplication before addition. Don't skip steps.
- Not distributing: When evaluating f(x + 3), you must distribute: 5(x + 3), not just 5x + 3.
Function Types Comparison
Here's a quick reference for evaluating different function types:
| Function Type | Example | Key Rule |
|---|---|---|
| Linear | f(x) = 3x + 7 | Multiply, then add |
| Quadratic | f(x) = x² - 5 | Solve exponents before adding |
| Polynomial | f(x) = 2x³ + x² - 3x + 1 | Work term by term |
| Rational | f(x) = (x + 2)/(x - 4) | Simplify numerator and denominator separately |
| Radical | f(x) = √(x + 9) | Add inside the root first |
| Constant | f(x) = 12 | Answer is always the constant |
Evaluating Composite Functions
A composite function is one function inside another. f(g(x)) means: first find g(x), then plug that result into f.
Example: If f(x) = x + 4 and g(x) = 2x, find f(g(3)).
Step 1: g(3) = 2(3) = 6
Step 2: f(6) = 6 + 4 = 10
Practice: Composite Functions
11. If f(x) = x² and g(x) = x + 1, find f(g(2)).
12. If f(x) = 3x - 2 and g(x) = x/3, find g(f(4)).
13. If f(x) = √x and g(x) = x² + 1, find f(g(3)).
Answers:
11. g(2) = 3, f(3) = 9 → 9
12. f(4) = 10, g(10) = 10/3 → 10/3
13. g(3) = 10, f(10) = √10 → √10
Tips That Actually Help
- Write out every step. Mental math invites errors.
- Use parentheses aggressively. They keep negatives and expressions organized.
- Check your work by plugging your answer back in. If f(3) = 11, verify: does f(3) = 2(3) + 5? Yes. ✓
- For composite functions, work inside-out. Always.
When You're Stuck
If you get confused evaluating f(x + h) or similar expressions, remember: you're not "solving" for anything. You're just simplifying an expression with a specific value substituted in.
The goal is algebraic simplification, not finding x. There's no x to find here.
Bottom Line
Evaluating functions is substitution followed by arithmetic. That's the entire process. The practice problems above cover the variations you'll encounter. Work through them until you can do them without hesitation.
If you got 10/13 correct on your first attempt through the exercises, you're ready to move on. If not, redo the ones you missed. There's no shortcut that replaces repetition.