How to Solve: Functions Basics

How to Solve: Functions Basics

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"Functions are the secret code of algebra—if you can read them, you can predict stock markets, design video games, and even pass your exam with top marks. Let’s crack the code."

What You Need To Know First

1. Variables & Expressions – You must know what x and y represent and how to substitute values.
2. Ordered Pairs – Understand how points like (2, 5) work on a graph.
3. Basic Equations – You should be comfortable solving y = 2x + 3 for y when x = 4.

Key Vocabulary

Formulas To Know

1. Function Notation
2. Formula: f(x) = [expression]
3. Variables:
   • f = name of the function (can be g, h, etc.)
   • x = input
   • [expression] = rule (e.g., 2x + 1)

4. MEMORISE THIS – You’ll use it in every function question.

5. Evaluating a Function

6. Formula: f(a) = [expression with x replaced by a]
7. Example: If f(x) = 3x – 2, then f(4) = 3(4) – 2 = 10.

8. MEMORISE THIS – Exams test this constantly.

9. Domain Restrictions

10. Denominator ≠ 0: If f(x) = 1/(x–3), domain is x ≠ 3.
11. Square Roots ≥ 0: If f(x) = √(x+2), domain is x ≥ –2.
12. Given on exam sheet (but you must know how to apply them).

Step-by-Step Method

How to Solve Any Function Problem

1. Identify the function rule – Look for f(x) = … or a table/graph.
2. Find what’s being asked – Is it f(5)? The domain? A graph?
3. Substitute (if evaluating) – Replace x with the given input.
4. Simplify – Do the math step by step.
5. Check for restrictions – Does the input break any rules (e.g., division by zero)?
6. Write the final answer – Include units or context if given.

Worked Example (Using the Steps)

Problem: Given f(x) = 2x² – 5x + 1, find f(–3).

1. Identify the function rule: f(x) = 2x² – 5x + 1
2. Find what’s being asked: Evaluate f(–3).
3. Substitute: Replace x with –3 → f(–3) = 2(–3)² – 5(–3) + 1
4. Simplify:
5. 2(9) + 15 + 1 (because (–3)² = 9 and –5 × –3 = +15)
6. 18 + 15 + 1 = 34
7. Check for restrictions: No division or square roots → all real numbers work.
8. Final answer: f(–3) = 34

Worked Examples

Example 1 – Basic: Evaluating a Function

Problem: If g(x) = 4x – 7, find g(2).

Solution: 1. Write the function: g(x) = 4x – 7 2. Substitute x = 2: g(2) = 4(2) – 7 3. Simplify: 8 – 7 = 1 4. Answer: g(2) = 1

What we did and why: We replaced x with 2 and followed the order of operations (PEMDAS). This is the most basic function skill—exams test it first.

Example 2 – Medium: Domain & Range

Problem: Find the domain and range of h(x) = √(x + 4).

Solution: 1. Domain:
- Square roots require x + 4 ≥ 0.
- Solve: x ≥ –4.
- Domain: x ≥ –4 (or [–4, ∞) in interval notation). 2. Range:
- √ gives non-negative outputs.
- Minimum output: h(–4) = √0 = 0.
- Range: y ≥ 0 (or [0, ∞)). 3. Answer:
- Domain: x ≥ –4
- Range: y ≥ 0

What we did and why: We used the rule that square roots can’t have negative numbers inside. The range comes from knowing √ always gives 0 or positive numbers.

Example 3 – Exam Style: Disguised Function

Problem: The cost C (in dollars) to rent a bike for t hours is given by C(t) = 15 + 3t. a) Find C(4) and interpret the answer. b) What is the domain if the shop is open for 10 hours?

Solution: a) Find C(4):
1. C(t) = 15 + 3t
2. Substitute t = 4: C(4) = 15 + 3(4) = 15 + 12 = 27
3. Interpretation: Renting for 4 hours costs $27. b) Domain:
- Shop is open for 10 hours → t can be from 0 to 10.
- Domain: 0 ≤ t ≤ 10

What we did and why: We treated C(t) like any other function but added real-world meaning. Exams love word problems—always write units and context.

Common Mistakes

Exam Traps

1-Minute Recap

(Spoken naturally, under 150 words)

"Alright, listen up—this is your 60-second functions cheat sheet. A function is just a rule that gives one output per input. Write it as f(x) = …, plug in the number, and simplify. Domain? All possible x values—watch for square roots (can’t be negative) and denominators (can’t be zero). Range? All possible y values—think about the lowest and highest outputs. Graphs? Use the vertical line test—if a line hits twice, it’s not a function. Exams will trick you with word problems, so underline the function rule and what’s being asked. And always check for restrictions before substituting. You’ve got this—go ace that test!