To learn Functions and Function Notation quickly and correctly, think of a function as a rule that gives exactly one output for each allowed input. The symbol f(x) is simply a compact way of saying “the output of function f when the input is x”. You already know how to substitute numbers into algebraic expressions; function notation makes that operation easy to describe and extend to tables, graphs and real-life relationships. These eduKateSG Mathematics Quick Notes teach the meaning of f(x), how to evaluate it and how to avoid common mistakes.
- Function: every permitted input has exactly one output.
- Read f(x): “f of x”, not f multiplied by x.
- Evaluate: replace x with the stated input using brackets when needed.
- Solve: if asked f(x) = a value, form and solve an equation.
- Respect the domain: exclude inputs for which the rule is not defined.
- Check: substitute each candidate input into the original function.
First, learn to translate a rule from words into function notation. Suppose each pencil costs $2 and there is a fixed $3 delivery charge. If x is the number of pencils, the total cost is f(x) = 2x + 3. For four pencils, f(4) = 2(4) + 3 = 11. The input is 4 pencils, and the output is $11. Notice the fixed charge makes this relationship linear but not directly proportional to x. Function notation gives the relationship a name without changing what the calculation means.
Second, keep the input in brackets when you substitute. Let f(x) = x² − 1. Then f(3) = 3² − 1 = 8, while f(−3) = (−3)² − 1 = 8. This function gives the same output for two different inputs, which is allowed. The rule requires one output per input, not one unique input per output. Now try f(0) = 0² − 1 = −1. Comparing these three evaluations helps you see how a function can be represented as a table or graph.
Third, distinguish evaluating a function from solving an equation about it. If f(x) = x² − 1, the instruction “find f(4)” asks you to put 4 into the rule: f(4) = 16 − 1 = 15. The instruction “solve f(x) = 8” asks which inputs give the output 8. Solve x² − 1 = 8, so x² = 9, giving x = 3 or x = −3. Both must be checked. The first task finds an output from an input; the second searches for inputs that give a particular output.
Worked example: function substitution
Given g(x) = 3x − 5, calculate g(6) and g(−2). For g(6), replace x with 6: 3 × 6 − 5 = 13. For g(−2), use brackets: 3(−2) − 5 = −6 − 5 = −11. You can write a short input-output table: input 6 → output 13, input −2 → output −11. The same rule works for every allowed input.
Worked example: find an input from an output
Given f(x) = 4x + 7, solve f(x) = 31. This means 4x + 7 = 31. Subtract 7 from both sides: 4x = 24. Divide by 4: x = 6. Check: f(6) = 4(6) + 7 = 31. The answer is an input, not the output 31. The question wording tells you which quantity to state.
Domain and range: which values are allowed?
The domain is the collection of permitted inputs. The range is the collection of outputs produced by those inputs. For f(x) = x² with all real x allowed, the domain is all real numbers, but the range is y ≥ 0 because a square is never negative. For f(x) = 1/(x − 2), x = 2 is not allowed because it would cause division by zero. Its real domain excludes 2, and its range excludes zero. Always check whether the problem gives its own restricted domain, such as positive whole numbers representing the number of tickets.
A square root example with a restricted domain
Let h(x) = √(x + 1) and work with real numbers. The expression under a real square root must be non-negative, so x + 1 ≥ 0, giving x ≥ −1. That is the domain. The principal square root itself is non-negative, so the range is h(x) ≥ 0. For example, h(3) = √4 = 2, while h(−2) is not a real number. A restricted domain is a mathematical condition, not an optional warning to ignore.
Composite functions: one rule followed by another
Suppose f(x) = 2x + 1 and g(x) = x². The expression f(g(2)) means evaluate the inner function first: g(2) = 2² = 4. Then f(4) = 2(4) + 1 = 9. But g(f(2)) gives f(2) = 5, then g(5) = 25. The order matters: 9 is not 25. Write arrows if the nested brackets look difficult: input 2 → g → 4 → f → 9.
Functions, equations and graphs are connected
If a graph uses x horizontally and y vertically, a function y = f(x) gives the height for each allowed x. A vertical line at a permitted input must meet the graph at exactly one point if it represents a function. For example, y = x² is a function because each x has one squared output, even though a horizontal line such as y = 4 meets the parabola at two inputs. This is the vertical line test and explains why repeated outputs do not violate the function rule.
Why a circle is not y as a function of x over its whole outline
The equation x² + y² = 25 draws a circle centred at the origin. At x = 0, there are two y-values, +5 and −5. So the complete circle is not a single-valued function y = f(x), because one input x = 0 has two outputs. But its upper semicircle y = √(25 − x²), on the domain −5 ≤ x ≤ 5, is a function. It produces one non-negative output for each allowed x. This is a useful reminder that a graph must pass the function test, not merely have an equation.
Common mistakes and easy corrections
- Reading f(x) as f × x: f is a function name; f(x) means its output.
- Writing f(−3) = −3² − 1: put the negative input inside brackets before squaring.
- Confusing f(5) with solving f(x) = 5: one gives an output; the other searches for inputs.
- Assuming outputs cannot repeat: a function may give the same output for different inputs.
- Ignoring restrictions: division by zero and negative real square-root inputs are not permitted.
- Doing composite functions in the wrong order: evaluate the innermost expression first.
Mini-test: calculate and explain
- For f(x) = 2x − 3, find f(5).
- For g(x) = x² + 2, find g(−4).
- For f(x) = 2x − 3, solve f(x) = 11.
- For h(x) = 1/(x + 4), state which real input is excluded.
- If f(x) = x + 1 and g(x) = 3x, find f(g(2)).
Answers with the working
1. f(5) = 10 − 3 = 7. 2. g(−4) = (−4)² + 2 = 18. 3. 2x − 3 = 11, so 2x = 14 and x = 7. 4. x = −4 is excluded because the denominator would be zero. 5. g(2) = 6, then f(6) = 7.
Your ten-minute function study routine
- Minutes 1–2: explain input, output and the meaning of f(x).
- Minutes 3–4: substitute one positive and one negative input.
- Minutes 5–6: solve an equation in the form f(x) = a number.
- Minutes 7–8: check a function’s domain and one composite evaluation.
- Minutes 9–10: verify your answers and explain the difference between evaluating and solving.
The real learning target is simple: you should be able to read a function rule, put an input into it accurately and explain how the output is produced. Once that habit is stable, more advanced functions and graphs become natural extensions of familiar algebra.
Connect this lesson to the next Quick Notes
Continue with Algebra · Linear Equations · Quadratic Graphs · Number Patterns and Sequences. Keep each study session small: explain, solve, check and revisit tomorrow.
For the wider learning approach, see Secondary 1 Mathematics Tutor Clementi | Small Groups Tutorials. The original reference article remains unchanged.