1st Year Math Chapter 2 Exercise 2.1 Notes: Functions (Punjab Board 2026-27)
Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.
Chapter 2, Functions and Graphs, follows Complex Numbers in the 14-unit Mathematics 11 (PECTAA) textbook. Exercise 2.1 introduces the formal definition of a function, along with domain, range, function notation, and the vertical line test.
What Does This Exercise Cover?
This exercise builds the foundation for the whole chapter: recognizing what makes a relation a function, describing its domain and range, and evaluating a function at a given input value.
Key Concepts
1. What Is a Function?
A function is a rule that assigns each input value exactly one output value. It can be thought of as a computing machine: an input x goes in, the function processes it, and exactly one output f(x) comes out. If any input produced more than one output, the relation would not be a function.
2. Domain and Range
The domain of a function is the complete set of possible input values (x-values). The range is the complete set of resulting output values (f(x)-values) produced by the function.
3. Function Notation
Functions are usually named with letters like f, g, or h, and written as f(x), read as ‘f of x’. This notation shows both the name of the function and the input it’s being evaluated at.
4. The Vertical Line Test
The vertical line test is a way to check, from a graph, whether a relation is a function: if any vertical line crosses the graph at more than one point, the relation is not a function, because that x-value would correspond to more than one output.
5. Evaluating a Function
To evaluate a function at a specific input, substitute that value everywhere the variable appears in the function’s rule, then simplify.
Solved Examples
Example 1: If f(x) = x² + 2x − 1, find f(3).
Substitute x = 3 into the rule: f(3) = (3)² + 2(3) − 1.
Simplify: f(3) = 9 + 6 − 1.
Answer: f(3) = 14
Example 2: Find the domain and range of the relation {(1, 2), (2, 4), (3, 6)}, and state whether it is a function.
The domain is the set of all first coordinates: {1, 2, 3}.
The range is the set of all second coordinates: {2, 4, 6}.
Since each input (1, 2, 3) is paired with exactly one output, this relation is a function.
Answer: Domain = {1, 2, 3}, Range = {2, 4, 6}; it is a function
Example 3: Explain why x = y² does not define y as a function of x.
Solving for y gives y = ±√x, meaning most positive x-values correspond to two different y-values.
For example, at x = 4, both y = 2 and y = −2 satisfy the equation.
Since one input (x = 4) produces two outputs, this fails the definition of a function.
Answer: x = y² does not define y as a function of x, since some x-values give two y-values
Sample MCQs
1. A function assigns each input to:
a) Many outputs b) Exactly one output c) No output d) Exactly two outputs
Answer: b) Exactly one output
2. The set of all possible input values of a function is called its:
a) Range b) Domain c) Codomain image d) Output set
Answer: b) Domain
3. If f(x) = 2x + 3, then f(4) equals:
a) 7 b) 8 c) 11 d) 14
Answer: c) 11
4. The vertical line test is used to determine whether a graph represents:
a) A straight line b) A function c) An inequality d) A constant
Answer: b) A function
5. The set of all output values produced by a function is called its:
a) Domain b) Range c) Input set d) Rule
Answer: b) Range
Important Short Questions
- Define a function using the idea of input and output.
- What is the difference between the domain and range of a function?
- Explain the vertical line test in your own words.
- If f(x) = x² − 4, find f(−2).
- Determine whether {(1, 3), (1, 5), (2, 7)} represents a function, and explain why or why not.
Important Long Questions
- Given f(x) = 3x² − 2x + 1, find f(0), f(1), and f(−2), showing all working.
- Explain what a function is, and describe how the vertical line test is used to check whether a graph represents one.
- Find the domain and range of the relation {(−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4)}, and state whether it defines a function.
- Explain, with an example, why x = y² does not define y as a function of x.
How to Approach This Exercise Effectively
- Always check for repeated x-values with different y-values first — that’s the fastest way to rule out a function.
- When evaluating f(a), substitute carefully into every occurrence of x, including inside powers.
- Practice listing the domain and range separately, in set notation, so the two are never confused.
- Sketch a rough graph whenever you’re unsure whether a relation is a function, and apply the vertical line test visually.
- Get comfortable with function notation early — f(x), g(x), and h(x) all follow the exact same evaluation process.
FAQs
Q: What is the difference between a relation and a function?
A: Every function is a relation, but not every relation is a function — a function specifically requires that each input maps to only one output.
Q: Why does the vertical line test work?
A: A vertical line represents a single fixed x-value; if it touches the graph more than once, that x-value has more than one corresponding y-value, which violates the definition of a function.
Q: Can two different inputs give the same output?
A: Yes — this is allowed. A function only requires that each input has exactly one output; it doesn’t restrict multiple inputs from sharing the same output.
Q: What’s the easiest way to check if a set of ordered pairs is a function?
A: Scan the first coordinates (x-values) for repeats — if the same x-value appears with two different y-values, it is not a function.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 2: Functions and Graphs, Exercise 2.1.