Exam-Style Questions.

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

$$ \int_{-2}^{2} \left( f(|x|) + f(x) \right) \, dx. $$

Worked Solution

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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 \( x \in \mathbb{R} \), \( x \neq \pm1\)

Worked Solution

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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 \( f(x) \lt g(x) \).

Worked Solution

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