IBDP Maths SL 2.11 Transformations of graphs AA HL Paper 1- Exam Style Questions- New Syllabus
Question
Most-appropriate topic codes (IB Mathematics Analysis and Approaches 2021):
SL 2.2: Concept of domain and range — part (c)
SL 2.11: Transformations of graphs (reflections) — part (c)
AHL 2.13: Rational functions of the form $f(x) = \frac{ax+b}{cx+d}$ — part (b)
▶️ Answer/Explanation
(a)
Substitute $x = 0$ into the expression for $f$:
$f(0) = \frac{3(0) – 2}{2(0) + 1} = \frac{-2}{1} = -2$
Result: $-2$
(b)
For a rational function of the form $\frac{ax+b}{cx+d}$, the horizontal asymptote is found by the ratio of the leading coefficients as $x \to \infty$[cite: 1007]:
$y = \frac{3}{2}$
Result: $y = \frac{3}{2}$ (or $y = 1.5$)
(c)
First, find the range of $f(x)$ for $x \geq 0$.
As $x \to \infty$, $f(x) \to \frac{3}{2}$ from below. At $x = 0$, $f(0) = -2$.
So the range of $f$ for $x \geq 0$ is $[-2, 1.5)$.
Since $g(x) = -f(x)$, this represents a reflection in the $x$-axis[cite: 1033].
Applying the reflection to the range: $[-1 \times -2, -1 \times 1.5) \implies [2, -1.5)$.
Reordering for the range of $g$: $-1.5 < y \leq 2$.
Result: $-\frac{3}{2} < g(x) \leq 2$