IB Mathematics SL 2.2 Concept of a function, domain, range and graph AI SL Paper 1- Exam Style Questions- New Syllabus
Question
The discrete function \( f \) is defined for the domain \( \{0, 1, 2, 3, 4\} \). The table below represents the mapping of \( x \) to \( f(x) \):
| \( x \) | \( 0 \) | \( 1 \) | \( 2 \) | \( 3 \) | \( 4 \) |
|---|---|---|---|---|---|
| \( f(x) \) | \( 3 \) | \( 1 \) | \( 0 \) | \( 4 \) | \( 2 \) |
(a) Find the value of \( x \) such that \( f(x) = 4 \).
(b) Determine the value of \( x \) for which the output is equal to the input.
(c) The inverse function of \( f \) is denoted by \( f^{-1} \). Copy and complete the table for the inverse mapping:
| \( x \) | \( 0 \) | \( 1 \) | \( 2 \) | \( 3 \) | \( 4 \) |
|---|---|---|---|---|---|
| \( f^{-1}(x) \) |
Most-appropriate topic codes (IB Math AI 2025):
• SL 2.2: Identity function and solution of \( f(x) = g(x) \) —Part (a), part (b)
• SL 2.2: Concept of an inverse function and its tabular representation — part (c)
• SL 2.2: Concept of an inverse function and its tabular representation — part (c)
▶️ Answer/Explanation
(a)
From the table, find the \( x \) value where \( f(x) = 4 \). This occurs when \( x = 3 \).
\(\boxed{3}\)
(b)
Find \( x \) such that \( f(x) = x \). From the table:
When \( x = 0 \), \( f(0) = 3 \) (not equal)
When \( x = 1 \), \( f(1) = 1 \) (equal)
No other \( x \) satisfies \( f(x) = x \).
\(\boxed{1}\)
(c)
The inverse function \( f^{-1}(x) \) gives the input \( x \) that produces the given output. From the original table:
\( f(2) = 0 \) ⇒ \( f^{-1}(0) = 2 \)
\( f(1) = 1 \) ⇒ \( f^{-1}(1) = 1 \)
\( f(4) = 2 \) ⇒ \( f^{-1}(2) = 4 \)
\( f(0) = 3 \) ⇒ \( f^{-1}(3) = 0 \)
\( f(3) = 4 \) ⇒ \( f^{-1}(4) = 3 \)
Completed table:
| \( x \) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \( f^{-1}(x) \) | 2 | 1 | 4 | 0 | 3 |