<p>It is given that</p>
<p style="text-align: center">$\mathrm{f}(x)=\left\{ \begin{array}{*{35}{l}} \sqrt{1-\frac{{{x}^{2}}}{{{a}^{2}}}} & \mathrm{for }-a\le x\le a, \\ 0 & \mathrm{for }a < x < 2a,\\\end{array}\right.$</p>
<p>and that $\mathrm{f}(x+3a)=\mathrm{f}(x)$ for all real values of $x$, where $a$ is a real constant.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y=\mathrm{f}(x)$ for $-4a\le x\le 6a$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">State the range of $\mathrm{f}$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain whether ${{\mathrm{f}}^{-1}}$ exists for $-a\le x\le 2a$.</span></div></li></ol>

$\mathrm{f}(x)=\left\{ \begin{array}{*{35}{l}} \sqrt{1-\frac{{{x}^{2}}}{{{a}^{2}}}} & \mathrm{for }-a\le x\le a, \\ 0 & \mathrm{for }a < x < 2a,\\\end{array}\right.$

and that $\mathrm{f}(x+3a)=\mathrm{f}(x)$ for all real values of $x$, where $a$ is a real constant.

Sketch the graph of $y=\mathrm{f}(x)$ for $-4a\le x\le 6a$.

Explain whether ${{\mathrm{f}}^{-1}}$ exists for $-a\le x\le 2a$.

Modulus Functions

Functions

International Baccalaureate Higher Level Mathematics

IB HL Math

N2013 P1 Q5 [Modified]

The function $y = \sqrt{1-\dfrac{x^{2}}{a^{2}}}$ for $-a \leq x \leq a$ is the upper half of the ellipse $\dfrac{x^{2}}{a^{2}}+y^{2}=1$ (a semicircle-like curve from $(-a,0)$ to $(a,0)$ with maximum $(0,1)$).

For $a < x < 2a$, $\mathrm{f}(x) = 0$.

Since $\mathrm{f}(x+3a) = \mathrm{f}(x)$, the function is periodic with period $3a$.

The graph consists of repeated semicircular arcs centred at $x = 0, \pm 3a, \pm 6a, \ldots$, each reaching height 1, with flat sections at $y=0$ between them.

Key points: $(-3a,1)$, $(0,1)$, $(3a,1)$, $(6a,1)$ at the peaks.

The line $y = \dfrac{1}{2}$ cuts the graph of $\mathrm{f}$ twice for $-a \leq x \leq 2a$.

Thus $\mathrm{f}$ is not one-one for $-a \leq x \leq 2a$.

Hence $\mathrm{f}^{-1}$ does not exist for $-a \leq x \leq 2a$.

N2013 P1 Q5 [Modified]

N2013 P1 Q5 [Modified]

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