For Integrated Programme students: Your current school materials, teacher instructions, and assessment scope take precedence because IP topic sequence and depth vary by school. This is an Eclat IP guide, not the O-Level / SEC G3 exam-track guide.
How this chapter applies
Eclat core: function notation, linear and quadratic graphs, power and exponential graph recognition, and graphical change form the main Eclat route.
School-sensitive extension: composite functions, inverse functions, and general transformation notation are school-sensitive IP depth beyond K310.
2027 national comparison: K310 Topic N6 includes ordered pairs, linear and quadratic functions, selected power functions, exponential functions, and estimating a curve gradient with a tangent. It does not require inverse or composite functions.
Check your school: confirm which graph families, transformation notation, and inverse or composite operations are assessed.
Q: What does IP EMaths Notes (Upper Sec, Year 3-4): 08) Functions and Transformations cover? A: Interpret function notation, compose mappings, and relate algebraic changes to graph translations or stretches.
The core idea is simple: A function is an input-output rule; a transformation tells how its graph moves or stretches.
Use it as a working check: For composites, work from the inside function outward. For graph shifts, changes outside f(x) move the graph vertically, while changes inside the brackets move it horizontally.
Then go one layer deeper: Use the examples to track each step clearly: substitute inputs, compose functions in the stated order, reverse operations for an inverse, and describe graph movement in plain words.
Function language appears in calculator prompts, inverse-relationship questions, and graph sketching. Track how algebraic edits move or stretch the curve.
K310 also expects students to recognise selected power graphs, exponential graphs, and a curve's gradient estimated from a drawn tangent. Composite functions, inverse functions, and general transformation notation remain useful IP depth, but they are not required by K310 merely because they appear in this chapter.
K310 graph-family checkpoint
Classify the graph before describing a transformation. Power and exponential functions have different shapes and should not be forced into the quadratic template.
Family
Shape or feature to recognise
Working check
Common trap
Negative even power
Two branches with both axes as asymptotes when the coefficient is positive.
The sign stays the same on both sides of the y-axis.
Drawing it as a straight line because the exponent is negative.
Reciprocal function
Branches in opposite quadrants when the coefficient is positive.
Check the signs of both coordinates.
Joining the branches across an asymptote.
Positive cubic
An increasing S-shaped curve when the coefficient is positive.
It passes through the origin.
Sketching a parabola.
Exponential function
Growth for a base greater than one, with repeated multiplication over equal x-steps.
Identify the y-intercept and horizontal asymptote.
Treating equal x-steps as equal y-increases.
To estimate the gradient of a curve at a point, draw a tangent that touches the curve locally. Choose two well-separated points on the tangent, not two points on the original curve, and calculate vertical change divided by horizontal change.
f(x) gives the output when input x is substituted.
Composite functions: (f∘g)(x)=f(g(x)).
Common transformations:
y=f(x)+k: shift up by k.
y=f(x−a): shift right by a.
y=af(x): vertical stretch by factor a.
y=f(ax): horizontal compression by factor a.
Transformation reading checkpoint
Before describing a graph transformation, locate where the change is made.
Algebraic change
Where it acts
Graph effect
Trap to avoid
f(x)+k
Outside the function output
Move every point up by k.
Do not change the input values.
f(x)−k
Outside the function output
Move every point down by k.
Do not call this a horizontal shift.
f(x−a)
Inside the input bracket
Move the graph right by a.
The sign feels reversed because the input must be larger to give the same bracket value.
f(x+a)
Inside the input bracket
Move the graph left by a.
Do not read the plus sign as "right".
af(x)
Outside the function output
Stretch vertical distances by factor a.
This changes y-values, not x-values.
f(ax)
Inside the input bracket
Compress horizontal distances by factor a, for a>1.
This changes x-values, not y-values.
Misconception check: outside changes affect outputs, so they move or stretch vertically. Inside changes affect inputs, so they move or stretch horizontally.
Inside multiplier checkpoint
For changes inside the bracket, keep the output value fixed and ask which new input gives the same old input.
New graph
Input condition for the same output
Point movement
Common trap
y=f(2x)
Need 2x=4 to use the old input 4.
A point (4,7) moves to (2,7).
Multiplying the x-coordinate by 2 instead of dividing it by 2.
y=f(x/2)
Need x/2=4 to use the old input 4.
A point (4,7)
y=f(−x)
Need −x=4 to use the old input 4.
A point (4,7)
Worked check: if (4,7) lies on y=f(x), then f(4)=7. On y=f(2x), set 2x=4, so x=2 and the matching point is (2,7). On y=f(x/2), set x/2=4, so x=8 and the matching point is (8,7).
Misconception check: inside multipliers change x-values in the reciprocal direction. The y-value stays 7 in the worked check because the output of f has not been multiplied.
Composite function order checkpoint
For composite notation, read the function closest to x first. Then feed that output into the outside function.
Expression
First step
Second step
Common trap
(f∘g)(x)
Find g(x).
Substitute the result into f.
Doing f(x) first because f is written on the left.
(g∘f)(x)
Find f(x).
Substitute the result into g.
Assuming (f∘g)(x)
f(g(2))
Evaluate g(2).
Put that number into f.
Substituting 2 into both functions separately.
Worked check: if f(x)=2x−3 and g(x)=x2+1, then
(f∘g)(x)=f(g(x))=2(x2+1)−3=2x2−1.
Misconception check: the composition symbol does not mean multiplication. (f∘g)(x) means "do g, then do f", not f(x)×g(x).
Worked example - Transformation tracker
The graph of y=x2 is transformed to y=3(x−2)2+5. Describe the sequence of transformations.
Start with y=x2.
x↦x−2 translates the graph 2 units to the right.
Multiplying by 3 stretches it vertically by factor 3 (parabola becomes narrower).
Adding 5 shifts the graph up by 5 units.
So the sequence is: translate right by 2, stretch vertically by 3, then shift up by 5.
Worked example - Compose functions carefully
Let f(x)=2x−3 and g(x)=x2+1. Find (f∘g)(x) and evaluate (g∘f)(2).
Compute (f∘g)(x)=f(g(x)):
First find g(x): x2+1.
Substitute into f: f(g(x))=2(x2+1)−3=2x2+2−3=2x2−1
Compute (g∘f)(2)=g(f(2)):
f(2)=2(2)−3=1
Thus (f∘g)(x)=2x2−1 and (g∘f)(2)=2.
Worked example - Find an inverse function
Given h(x)=23x−5, determine h−1(x) and state the transformation that links h and h−1.
Let y=23x−5.
Swap x and y: x=23y−5. This reflects that an inverse reverses the original mapping-the output y from h becomes the input for h−1, while the original input x becomes the output we now solve for.
Solve for y:
Multiply by 2: 2x=3y−5.
Add 5: 3y=2x+5.
Divide by 3: y=32x+5
Therefore h−1(x)=32x+5.
The original function multiplies by 3, subtracts 5, then divides by 2. The inverse reverses these steps: multiply by 2, add 5, then divide by 3. Graphically, h and h−1 are reflections of each other across the line y=x.
Inverse function check checkpoint
After finding an inverse, check that it reverses the original mapping. This catches sign errors that still look algebraically neat.
Check
What to do
What it proves
Trap to avoid
Operation check
List the operations in f, then reverse them in the opposite order for f−1.
The inverse undoes each step.
Reversing the operations but keeping them in the original order.
Composition check
Compute f(f−1(x)) or f−1(f(x)).
A correct inverse gives back x.
Stopping after solving for y without checking the result.
Graph check
Compare the graphs of y=f(x) and y=f−1(x).
They should reflect in the line y=x
Worked check: for h(x)=23x−5 and h−1(x)=32x+5, substitute back:
h(h−1(x))=23(32x+5)−5=22x+5−5=x.
Misconception check: an inverse function is not the same as a reciprocal. f−1(x) reverses the mapping; it does not mean f(x)1.
Practice Quiz
Consolidate function evaluation, compositions, and transformation language with auto-marked prompts.
Try this
Given f(x)=2x−1 and g(x)=x2+3, find (g∘f)(x) and state the value when x=−2.
