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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | IB Analysis and Approaches |
The graph of \( y = f(|x|) \) for \( -2 \leq x \leq 2 \) is shown in the following diagram.
(a) On the following axes, sketch the graph of \( y = |f(|x|)| \) for \( -2 \leq x \leq 2 \).
It is given that \( f \) is an odd function.
(b) On the following axes, sketch the graph of \( y = f(x) \) for \( -2 \leq x \leq 2 \).
It is also given that
$$ \int_{0}^{2} f(|x|) \, dx = \dfrac{2}{3} $$(c) Write down the value of
$$ \int_{-2}^{2} f(x) \, dx; $$(d) Evaluate
$$ \int_{-2}^{2} \left( f(|x|) + f(x) \right) \, dx. $$2. | IB Analysis and Approaches |
A function \( f \) is defined by \( f(x) = \arccos\left(\frac{x^2}{1-x^4}\right) \), \( x \in \mathbb{R} \), \( 0.785 \lt |x| \lt 5\).
(a) Show that \( f \) is an even function.
(b) Find the equations of the horizontal asymptote of \( y = f(x) \).
(c) Find \( f'(x) \) for \( x \in \mathbb{R} \), \( x \neq \pm1 \).
(d) Determine whether \( f \) is increasing or decreasing for \( x > 1 \).
(e) Find an expression for \( f^{-1}(x) \), for \( x \in \mathbb{R} \), \( x \neq \pm1\)
3. | IB Analysis and Approaches |
Consider the function \(f(x) = \dfrac{x^3+2x}{5}, \; x \in \mathbb{R}\).
(a) Show that \(f\) is an odd function.
The function \(g\) is given by:
$$ g(x) = \dfrac{3x-3}{x^2+x-2} \quad \text{ where } x \in \mathbb{R}, x \neq 1, \; x \neq -2.$$ (b) Solve the inequality \( f(x) \lt g(x) \).If you would like space on the right of the question to write out the solution try this Thinning Feature. It will collapse the text into the left half of your screen but large diagrams will remain unchanged.
The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.
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