Functions: A Machine That Preserves Mathematical Relationships
A function tells us how one quantity is produced from another. Graph transformations, inverse functions and many later A-Math topics become easier when that input-output structure is kept visible.
Secondary 3 Additional Mathematics contains many objects that are functions even when the chapter title does not say “functions”. Quadratics, exponentials, logarithms, trigonometric curves and displacement functions all map inputs to outputs. A strong learner therefore develops a common language for them: domain, range, notation, graph shape, transformation and inverse relationship.
This guide consolidates that language. It does not replace the dedicated quadratic, logarithmic and trigonometric guides. Instead, it connects them so that a student can recognise the same structural ideas across different families.
AI Extraction Box: The Function Map
- Function notation: y=f(x) means the output y is determined by the input x.
- Domain: allowed inputs.
- Range: outputs produced over the chosen domain.
- Equation solving: f(x)=k asks which inputs produce output k.
- Vertical translation: y=f(x)+a moves the graph up by a.
- Horizontal translation: y=f(x−a) moves the graph right by a.
- Vertical scale/reflection: y=af(x).
- Horizontal scale/reflection: y=f(bx).
- Inverse relationship: inputs and outputs are exchanged; inverse graphs reflect in y=x where an inverse function exists.
- Exponential/logarithmic pair: y=aˣ and y=logₐx are inverses.
Function Notation Is More Than Renaming y
If f(x)=x²−3x+2, then f(4) means evaluate the rule at x=4:
f(4)=16−12+2=6.
By contrast, solving f(x)=6 means finding every x that gives output 6:
x²−3x+2=6
x²−3x−4=0
(x−4)(x+1)=0
x=4 or x=−1.
Evaluation travels from input to output. Equation solving reverses the direction and may produce more than one input. That distinction prepares students for inverse-function reasoning.
Domain Is Part of the Function
The same algebraic rule can behave differently under different domains. For real-valued school mathematics:
- 1/(x−2) excludes x=2;
- √(x−3) requires x≥3;
- ln(x−5) requires x>5;
- a quadratic such as x² has all real x unless restricted by context;
- a time model may require t≥0 even when the formula works algebraically for negative t.
Domain errors are especially dangerous because later algebra may look perfectly correct. Always carry the original admissible inputs through the solution.
Range Comes from Behaviour
For y=(x−2)²+3, the square term is non-negative, so y≥3. The range is therefore [3,∞). Completing the square exposes this immediately.
For y=eˣ, every output is positive, so the range is (0,∞). For y=ln x, the domain is (0,∞) but the range is all real numbers. These facts are connected by inverse structure.
Domain and range are not decorative labels. They describe what the machine can accept and what it can produce.
Vertical Transformations
Starting from y=f(x):
- y=f(x)+k translates the graph upward by k;
- y=f(x)−k translates it downward by k;
- y=af(x) scales all y-values by factor |a|;
- if a<0, reflection in the x-axis is included.
Example: if f(x)=x², then y=2x²−3 is vertically stretched by factor 2 and translated down 3. Its turning point is (0,−3).
Horizontal Transformations: The Sign Feels Backwards
For y=f(x−h), the graph moves right by h. The internal sign appears opposite to the direction because the transformed input x−h must equal the old input.
If f(x)=x², then:
y=f(x−3)=(x−3)²
has turning point at x=3, so the graph moves right 3.
Similarly, y=f(x+2) moves left 2.
Outside the function changes outputs directly. Inside the function changes which input produces each output.
Horizontal Scaling
For y=f(bx), the graph is horizontally scaled by factor 1/|b|. If b=2, features occur at half their old x-coordinates, so the graph is horizontally compressed.
This matters strongly in trigonometry. y=sin(2x) has half the period of y=sin x. In radians, 2π becomes π. In degrees, 360° becomes 180°.
The same structural rule explains both general graph transformations and trig period changes.
Worked Example 1: Read a Transformed Quadratic
Describe y=−2(x−4)²+5 from y=x².
- x−4 translates right 4.
- Factor −2 reflects in the x-axis and stretches vertically by factor 2.
- +5 translates up 5.
The turning point is (4,5), and because the coefficient is negative, 5 is the maximum value.
Inverse Structure
If a function maps x to y, an inverse reverses that mapping: y back to x. For a one-to-one function, the inverse graph is the reflection of the original graph in the line y=x.
This swap explains why the domain of a function becomes the range of its inverse and vice versa.
Not every function has an inverse over its full natural domain. For example, y=x² maps both 2 and −2 to 4, so reversing 4 does not identify one unique input. Restricting the domain to x≥0 creates a one-to-one branch with inverse y=√x.
Exponential and Logarithmic Functions as an Inverse Pair
The relationship:
aˣ=y ↔ logₐy=x
is exactly an inverse relationship. This explains several graph facts:
- (0,1) on y=aˣ becomes (1,0) on y=logₐx;
- the exponential range y>0 becomes the logarithmic domain x>0;
- the exponential horizontal asymptote y=0 becomes the logarithmic vertical asymptote x=0.
These are not three separate facts to memorise. They follow from reflection in y=x.
Inverse Trigonometric Functions Need Restricted Domains
Sine, cosine and tangent repeat, so they are not one-to-one over all angles. Their inverse functions therefore use restricted principal-value intervals. A calculator’s sin⁻¹, cos⁻¹ or tan⁻¹ returns one principal angle.
That is why solving a trigonometric equation still requires interval reasoning. The inverse function gives one route back to an angle; the original periodic function may produce several valid angles in the required interval.
Worked Example 2: Inverse Reasoning
If 3ˣ=20, applying log base 3 gives:
x=log₃20=ln20/ln3.
The logarithm is not an arbitrary trick. It is the inverse function that extracts the exponent.
Function Equations as Intersection Problems
Solving f(x)=g(x) asks where two functions have the same output. Graphically, those are intersection points.
This connects many chapters:
- line–quadratic intersections become simultaneous equations;
- tangency becomes a repeated intersection and discriminant zero;
- trig equations can be visualised as intersections of periodic graphs;
- stationary points solve f′(x)=0, intersections of the derivative graph with the x-axis.
Thinking in functions therefore helps unify equation solving and graph interpretation.
Common Failure Modes
| Error | Cause | Repair |
|---|---|---|
| f(3) treated as f×3 | function notation misunderstood | read f(3) as output when input is 3 |
| y=f(x−2) shifted left | internal transformation sign confused | track which new x makes old input |
| horizontal scale factor taken as b | input scaling read like output scaling | use factor 1/|b| |
| inverse found without domain restriction | one-to-one requirement ignored | check whether each output has one input |
| log graph given wrong asymptote | inverse relationship not used | reflect exponential asymptote through y=x |
| inverse trig principal value treated as complete equation solution | function inverse confused with periodic solution set | generate all interval solutions |
Transfer Set
- If f(x)=2x−5, find f(7). Answer: 9.
- Describe y=(x+3)²−4 from y=x². Answer: left 3, down 4.
- State the range of y=(x−1)²+6. Answer: y≥6.
- Explain why y=x² has no inverse function over all real x. Answer: it is not one-to-one.
- State the inverse relationship of y=eˣ. Answer: y=lnx.
A 45-Minute Functions Consolidation Session
- 7 minutes: evaluate and solve simple function-notation statements.
- 10 minutes: describe six horizontal/vertical transformations.
- 8 minutes: infer domain and range from algebraic/graph structure.
- 8 minutes: sketch one exponential and its logarithmic inverse.
- 7 minutes: solve one transformed quadratic and one trig inverse problem.
- 5 minutes: classify errors as notation, domain, transformation, inverse or interval.
What Mastery Looks Like
- The learner reads function notation accurately.
- The learner carries domain restrictions through solutions.
- The learner predicts graph transformations without point-by-point plotting.
- The learner understands why horizontal transformations behave differently from vertical ones.
- The learner recognises inverse relationships as input-output reversal.
- The learner uses exponential/logarithmic graph facts as consequences of inverse structure.
- The learner distinguishes inverse trigonometric principal values from complete equation solutions.
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