Home / IB Mathematics AHL 2.7 Composite functions AI HL Paper 1- Exam Style Questions

IB Mathematics AHL 2.7 Composite functions AI HL Paper 1- Exam Style Questions

IB Mathematics AHL 2.7 Composite functions AI HL Paper 1- Exam Style Questions- New Syllabus

Question

The graph of \( y = f(x) \) for \( 0 \leq x \leq 10 \) is shown below. The graph intercepts the axes at \( (10, 0) \) and \( (0, 5) \).

Graph of y = f(x)

Part (a):
Write down the value of
(i) \( f(4) \)
(ii) \( f \circ f(4) \)
(iii) \( f^{-1}(3) \)

Part (b):
On the axes above, sketch the graph of \( y = f^{-1}(x) \). Show clearly where the graph intercepts the axes.

▶️ Answer/Explanation
Detailed Solutions

Part (a)

(i) To find \( f(4) \), locate \( x = 4 \) on the x-axis and trace vertically to the curve. The graph shows \( y \approx 1 \).

Answer: \( f(4) = 1 \)

(ii) To find \( f \circ f(4) \), first use \( f(4) = 1 \). Then, find \( f(1) \) by locating \( x = 1 \) on the x-axis and tracing vertically. The graph shows \( y \approx 3 \).

Answer: \( f \circ f(4) = f(1) = 3 \)

(iii) To find \( f^{-1}(3) \), find the \( x \)-value where \( f(x) = 3 \). Locate \( y = 3 \) on the y-axis and trace horizontally to the curve. The graph shows \( x \approx 1 \).

Answer: \( f^{-1}(3) = 1 \)

Part (b)

As per the markscheme, sketch the graph of \( y = f^{-1}(x) \):

– The graph is a concave up curve.

– It has a \( y \)-intercept at \( (0, 10) \).

– It has an \( x \)-intercept at \( (5, 0) \).

– The curve passes through either \( (2, 2) \) or both \( (1, 4) \) and \( (3, 1) \).

Graph of f(x) and f^-1(x)

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