IB Mathematics AA SL Composite functions Study Notes
IB Mathematics AA SL Composite functions Study Notes Offer a clear explanation of Composite functions , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Composite functions
Composite Functions
Introduction to Composite Functions
A composite function is a function that is formed by combining two or more functions. The output of one function becomes the input of the next. The notation for a composite function is written as \( (f \circ g)(x) = f(g(x)) \), meaning that function \( g(x) \) is applied first, followed by \( f(x) \).
Example of Composite Functions
- Let \( f(x) = x^2 \) and \( g(x) = x – 4 \). Then:
- \( f(g(x)) = (x – 4)^2 = x^2 – 8x + 16 \)
- \( g(f(8)) = g(64) = 60 \)
- \( g(g(3)) = g(-1) = -5 \)
Transformations of Graphs
Graph transformations alter the appearance and position of the graph of a function. These transformations include shifts, stretches, and reflections.
Horizontal Shifts
- f(x + a): Translation left by \( a \) units.
- f(x – a): Translation right by \( a \) units.
Vertical Shifts
- f(x) + b: Translation up by \( b \) units.
- f(x) – b: Translation down by \( b \) units.
Stretches
- f(ax): Horizontal stretch by a factor of \( 1/a \).
- f(x/a): Horizontal stretch by a factor of \( a \).
- af(x): Vertical stretch by a factor of \( a \).
- (1/a)f(x): Vertical stretch by a factor of \( 1/a \).
Reflections
- f(-x): Reflection in the \( y \)-axis.
- -f(x): Reflection in the \( x \)-axis.
Note: For reflections, only the \( x \)-values or \( f(x) \) are affected, not both simultaneously.
General Form of Transformations
The general form for transformations of a function is \( a f(b(x + c)) + d \), where:
- \( a \) controls vertical stretches and reflections.
- \( b \) controls horizontal stretches and reflections.
- \( c \) represents a horizontal shift.
- \( d \) represents a vertical shift.
Key Points
- Composite functions involve applying one function to the result of another.
- Graph transformations can change the shape and position of the graph.
- The general form allows for shifts, stretches, and reflections to be combined.
Example of Graph Transformation
For the function \( f(x) = x^2 \), consider the transformation \( f(2x + 1) \). Here’s how the translation happens:
- Translation of \( (0) \): Horizontal shift of \( -\frac{1}{2} \) units.
- Horizontal stretch by a factor of \( \frac{1}{2} \): The graph will be stretched horizontally by a factor of \( 2 \).
- For \( 2f(x) \): The \( x \)-values stay the same, but the \( y \)-coordinates will be doubled.
- For \( f(\frac{1}{2}x) \): The \( x \)-values will be halved, but the \( y \)-coordinates remain unchanged.
Example : Composite Function Calculation
Let \( f(x) = 3x + 2 \) and \( g(x) = x^2 – 1 \). Compute the following:
- f(g(x)): We substitute \( g(x) \) into \( f(x) \):
- \( f(g(x)) = 3(x^2 – 1) + 2 = 3x^2 – 3 + 2 = 3x^2 – 1 \)
- g(f(2)): First, evaluate \( f(2) = 3(2) + 2 = 6 + 2 = 8 \), then apply \( g(8) \):
- \( g(f(2)) = g(8) = 8^2 – 1 = 64 – 1 = 63 \)
- f(f(1)): First, evaluate \( f(1) = 3(1) + 2 = 3 + 2 = 5 \), then apply \( f(5) \):
- \( f(f(1)) = f(5) = 3(5) + 2 = 15 + 2 = 17 \)
IB Mathematics AA SL Composite Function Exam Style Worked Out Questions
Question
The graph of y = f (x) for -4 ≤ x ≤ 6 is shown in the following diagram.
(a) Write down the value of
(i) f (2) ;
(ii) ( f o f )(2) . [2]
(b) Let g(x) = \(\frac{1}{2} f (x) +1\) for -4 ≤ x ≤ 6 . On the axes above, sketch the graph of g . [3]
▶️Answer/Explanation
Ans:
(a) (i) f(2) = 6
(ii) (fof)2=− 2 [2 marks]
(b) 
Question
Let \(f(x) = 8x + 3\) and \(g(x) = 4x\), for \(x \in \mathbb{R}\).
Write down \(g(2)\).[1]
Find \((f \circ g)(x)\).[2]
Find \({f^{ – 1}}(x)\).[2]
▶️Answer/Explanation
Markscheme
\(g(2) = 8\) A1 N1
[1 mark]
attempt to form composite (in any order) (M1)
eg\(\,\,\,\,\,\)\(f(4x),{\text{ }}4 \times (8x + 3)\)
\((f \circ g)(x) = 32x + 3\) A1 N2
[2 marks]
interchanging \(x\) and \(y\) (may be seen at any time) (M1)
eg\(\,\,\,\,\,\)\(x = 8y + 3\)
\({f^{ – 1}}(x) = \frac{{x – 3}}{8}\,\,\,\,\,\left( {{\text{accept }}\frac{{x – 3}}{8},{\text{ }}y = \frac{{x – 3}}{8}} \right)\) A1 N2
[2 marks]