Secondary Mathematics · Worked Repair Guide 28
A function is not merely an equation with the letter f. It is a rule that assigns each allowed input exactly one output. The same relationship can be represented by a formula, table, mapping, ordered pairs or graph. The challenge is learning to keep the input-output structure stable while the representation changes.
This guide develops one central habit: state the allowed inputs before interpreting the outputs. Domain determines what may enter. The rule determines what happens. Range records what actually comes out. A formula can be algebraically neat while still being undefined at certain inputs.
The conceptual owners remain Functions and Graphs and Why Do Functions Feel So Abstract in Secondary Mathematics?. This page adds a worked-repair route without replacing the Additional Mathematics function-synthesis estate.
1. A function assigns one output to each allowed input
Let f(x)=3x−2. Input x=5 produces f(5)=15−2=13.
The notation f(5) means “the output of function f when the input is 5”. It does not mean f×5.
Entry check: f(−4)=3(−4)−2=−14. Brackets protect the negative input.
One rule can produce many outputs because different inputs can be used.
2. One input cannot have two different outputs in the same function
The mapping 1→2 and 1→3 does not define a function from the stated input set because input 1 has two outputs.
By contrast, 1→5 and 2→5 is allowed. Different inputs may share the same output.
This distinction separates “function” from “one-to-one function”. A function need not be one-to-one.
The rule is one output per allowed input, not one unique output value for the whole domain.
3. Domain is the set of allowed inputs
For f(x)=3x−2 over all real numbers, every real x can be substituted, so the natural real domain is all real numbers.
For h(x)=1/(x−4), x=4 is forbidden because the denominator becomes zero. The real domain is all real x except 4.
For g(x)=√(x+2) over the reals, require x+2≥0, so x≥−2.
Domain restrictions come from the rule or from the context, and they should be stated before solving or graphing.
4. Range is the set of outputs actually produced
For f(x)=x² with real domain, every output is non-negative, so range y≥0.
For f(x)=x² with restricted domain {−2,−1,0,1,2}, the range is {0,1,4}.
The range depends on both the rule and the domain.
Changing the allowed inputs can change the range even when the formula is unchanged.
5. Tables make input-output structure explicit
Suppose x-values 1,2,3 produce y-values 4,7,10. The outputs increase by 3 when inputs increase by 1.
A linear rule fitting these values is y=3x+1.
Check x=1 gives 4, x=2 gives 7 and x=3 gives10.
A table can suggest a rule, but a finite table does not logically prove a unique formula unless the function family or structure is specified.
6. Ordered pairs are points and function records at once
The ordered pair (a,b) on the graph y=f(x) means f(a)=b.
If (4,9) lies on the graph, then f(4)=9.
The first coordinate is the input, the second the output.
Swapping coordinates changes the statement to f(9)=4, which need not be true.
7. The vertical line test checks whether a graph represents a function of x
If any vertical line cuts a graph more than once, one x-value is paired with more than one y-value, so the graph is not a function y=f(x).
A circle fails this test because many vertical lines intersect it twice.
A sideways parabola can also fail, while an ordinary vertical parabola y=x² passes.
The test is graphical form of the “one output per input” definition.
8. Evaluate a function before trying to solve it
For f(x)=3x−2, f(5)=13 is evaluation: the input is known and the output is required.
Solving f(x)=19 is different. Write 3x−2=19, so 3x=21 and x=7.
Evaluation moves from input to output. Solving reverses the question and asks which input produces a target output.
Do not treat f(x) as an unknown multiplier; it names the output expression.
9. Roots are inputs that make the function value zero
For f(x)=2x+6, a root satisfies f(x)=0.
2x+6=0 gives x=−3.
Graphically, the point (−3,0) is the x-intercept.
The root is the input −3; the intercept is the coordinate pair (−3,0). State the object requested by the question.
10. The y-intercept is the output at x=0 when zero is in the domain
For f(x)=5x−7, f(0)=−7, so the y-intercept is (0,−7).
For h(x)=1/x, x=0 is not in the domain, so there is no y-intercept.
Do not substitute a forbidden input simply because the intercept routine normally uses zero.
Domain comes before intercept calculation.
11. A quadratic function has structural features beyond its roots
Consider f(x)=(x−4)²−9.
The squared term is smallest when x=4, so the turning point is (4,−9). Since the coefficient of the square is positive, the real range is y≥−9.
Roots satisfy (x−4)²=9, so x=1 or7.
The completed-square form reveals turning point and range more directly than an expanded form.
12. Many-to-one functions remain valid functions
For f(x)=x², f(2)=4 and f(−2)=4.
Two inputs share one output, so the function is many-to-one over the real domain.
This does not violate the function definition because neither input has two outputs.
It does affect whether a simple inverse function can be defined on the whole real domain; that is an extension owned more fully by the Additional Mathematics function route.
13. Discrete domains produce isolated graph points
Suppose n counts whole objects and C=2n+1 with domain n∈{1,2,3,4}.
The outputs are {3,5,7,9}. The graph consists of four isolated points.
Joining them with a continuous line would imply values such as n=2.5 are allowed, contradicting the stated domain.
Use the domain to decide whether a graph should be continuous or discrete.
14. Continuous models allow every input in an interval
A model for distance travelled over time may use every real t from 0 to10 seconds, producing a continuous domain interval.
A count of completed packages after each whole minute may instead be discrete.
The formula alone may look similar; the variable meaning determines the valid inputs.
This connects to the existing operating-manual route on Discrete and Continuous Models.
15. Reciprocal functions show how restrictions shape graph behaviour
For f(x)=6/x, domain excludes x=0.
At x=3, f(x)=2. At x=−3, f(x)=−2.
As x approaches zero from the positive side, outputs become large positive; from the negative side, large negative. The graph does not cross x=0.
This behaviour follows from division by values approaching zero and should not be “filled in” with a point at the forbidden input.
16. Square-root functions begin where the radicand becomes non-negative
For f(x)=√(x+2), the real domain is x≥−2.
The graph begins at (−2,0) and produces non-negative outputs, so range y≥0.
At x=7, f(7)=3. At x=−3, the real function is undefined.
A calculator error message at x=−3 is not random; it reflects the real-domain restriction.
17. Piecewise rules make the domain split visible
As an optional extension, define f(x)=x+2 for x<0 and f(x)=x² for x≥0.
At x=−3 use the first rule: f(−3)=−1. At x=2 use the second: f(2)=4. At x=0 use the second rule because its condition includes equality.
A piecewise function is still one function because every allowed input is assigned exactly one output.
Boundary symbols such as < and ≥ determine which rule owns the endpoint.
18. Function models can be valid only over a contextual domain
An invented cost model C(n)=8+2.5n may be meaningful only for non-negative quantities n.
Algebraically C(−4)=−2 is easy to calculate, but a negative number of kilometres or items may be meaningless in the stated context.
The model’s contextual domain can therefore be narrower than the formula’s natural algebraic domain.
Always return the mathematical answer to the quantity represented by the variable.
19. Capstone: read one function through several representations
Let f(x)=x²−6x+5.
Factor form: f(x)=(x−1)(x−5), so roots are 1 and5.
Completed-square form: f(x)=(x−3)²−4, so turning point is (3,−4) and real range is y≥−4.
At x=0, f(0)=5, so y-intercept is (0,5).
The formula, factors and completed square describe the same function while making different features visible.
20. Independent practice
- For f(x)=3x−2, find f(5).
- Find f(−4).
- Solve f(x)=19.
- For g(x)=x²+1, find g(−3).
- State the real domain of h(x)=1/(x−4).
- State the real domain of p(x)=√(x+2).
- State the real range of y=x².
- A mapping sends 1 to 2 and 1 to 3. Is it a function?
- A mapping sends 1 to5 and2 to5. Is it a function?
- Explain what the vertical line test checks.
- Find the root of y=2x+6.
- For y=(x−4)²−9, state the turning point and real range.
- Domain is {1,2,3}; f(x)=2x+1. Find the range.
- For f(x)=x/2+4, solve f(x)=0.
- For f(x)=6/x, find f(3) and state the real domain.
- A table gives x=1,2,3 and y=4,7,10. Give a linear rule fitting all three points.
- Explain why f(x)=x² is still a function although f(−2)=f(2).
- An input variable counts whole objects. Should its graph automatically be joined continuously? Explain.
- For domain {−2,−1,0,1,2} and f(x)=x², state the range.
- If point (a,b) lies on y=f(x), what function statement does it represent?
21. Worked answers
1. 13.
2. −14.
3. x=7.
4. 10.
5. All real x except 4.
6. x≥−2.
7. y≥0.
8. No. Input 1 has two outputs.
9. Yes. Many inputs may share one output.
10. Whether any x-value corresponds to more than one point on the graph; if a vertical line cuts more than once, the graph is not y as a function of x.
11. x=−3.
12. Turning point (4,−9); range y≥−9.
13. {3,5,7}.
14. x=−8.
15. f(3)=2; domain all real x except0.
16. y=3x+1.
17. Each individual input still has exactly one output; the function is many-to-one, not one-to-many.
18. No. If only whole counts are allowed, intermediate non-integer inputs may be invalid, so the graph should reflect the discrete domain.
19. {0,1,4}.
20. f(a)=b.
22. Diagnose function errors by separating input, rule and output
Common failures include treating f(x) as multiplication, forgetting a denominator or square-root restriction, confusing roots with intercept coordinates, claiming a many-to-one mapping is not a function, or drawing a continuous line through a discrete domain.
A useful correction note says “domain first”, “f(a)=output”, “root means f(x)=0”, “one output per input”, or “context restricts allowed x-values”.
Then switch representation: formula to table, table to graph, graph to verbal description. Function understanding becomes robust when the same input-output relationship survives the change.
23. Continue through the BTT learning routes
Return to the BTT Mathematics Hub or BTT Mathematical Lab. Use Graphs, Tables and Relationships for basic representations, Quadratic Equations, Factorisation and Roots for quadratic features, and Direct and Inverse Proportion, Variation and Rates for proportional function families.
Within Batch 07, continue to Sampling, Scatter Plots, Correlation and Lines of Best Fit for fitted functions, Trigonometry, Pythagoras, Bearings and Elevation, or Surface Area, Volume and Composite Solids.
24. Sources and scope
The mappings, tables, domains and practice questions are original teaching material. Function, domain and range terminology is used in its standard mathematical sense.
For the current Singapore Secondary curriculum doorway, see MOE: Curriculum for secondary schools. Match piecewise, reciprocal and square-root extensions to the learner’s actual subject level and school programme.