Given $f(x) = an^{-1} \left( \frac{1}{2} x \right)$ and $g(x) = \sin(x - 30^{\circ})$ for $x \in [-90^{\circ}; 180^{\circ}]$ 6.1 On the same set of axes draw the graphs of $f$ and $g$. Show clearly on your graphs the turning points and asymptotes, if any. 6.2 Write down the period of $f$. 6.3 For what values of $x$ is $f(x) \cdot g(x) < 0$ for $x \in [-90^{\circ}; 120^{\circ}]?$ 6.4 Write down the equation(s) of the asymptotes of $h \cdot h(x) = f(x + 10^{\circ})$ for $x \in [-90^{\circ}; 180^{\circ}]$.

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Given $f(x) = an^{-1} \left( \frac{1}{2} x \right)$ and $g(x) = \sin(x - 30^{\circ})$ for $x \in [-90^{\circ}; 180^{\circ}]$ 6.1 On the same set of axes draw the graphs of $f$ and $g$ - NSC Mathematics - Question 6 - 2017 - Paper 2

Question 6

$Given---$f(x)-=--an^{-1}-\left(-\frac{1}{2}-x-\right)$-and-$g(x)-=-\sin(x---30^{\circ})$-for-$x-\in-[-90^{\circ};-180^{\circ}]$----6.1-On-the-same-set-of-axes-draw-the-graphs-of-$f$-and-$g$-NSC Mathematics-Question 6-2017-Paper 2.png$

Given $f(x) = an^{-1} \left( \frac{1}{2} x \right)$ and $g(x) = \sin(x - 30^{\circ})$ for $x \in [-90^{\circ}; 180^{\circ}]$ 6.1 On the same set of axes draw t... show full transcript

Worked Solution & Example Answer:Given $f(x) = an^{-1} \left( \frac{1}{2} x \right)$ and $g(x) = \sin(x - 30^{\circ})$ for $x \in [-90^{\circ}; 180^{\circ}]$ 6.1 On the same set of axes draw the graphs of $f$ and $g$ - NSC Mathematics - Question 6 - 2017 - Paper 2

Step 1

6.1 On the same set of axes draw the graphs of $f$ and $g$

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Answer

To draw the graphs of $f(x) = \tan^{-1}\left(\frac{1}{2}x\right)$ and $g(x) = \sin(x - 30^{\circ})$ , we first need to determine their characteristics:

For $f(x)$ :
- As $x \to -\infty$ , $f(x)$ approaches $-90^{\circ}$ and as $x \to +\infty$ , it approaches $90^{\circ}$ .
- There are no vertical asymptotes.
- The function is continuous and smooth.
For $g(x)$ :
- The function has a period of $360^{\circ}$ .
- The critical points occur at $30^{\circ}$ plus multiples of $360^{\circ}$ in the sine cycle, leading to maxima and minima intervals.
- Vertical asymptotes do not exist.

We then plot these points to clearly indicate the turning points and behavior of both functions.

Step 2

6.2 Write down the period of $f$

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Answer

The function $f(x) = \tan^{-1}\left(\frac{1}{2}x\right)$ does not have a specific period defined as it approaches asymptotes gradually rather than returning to a repeating pattern. Hence, its period is infinite. However, if considering periodic extensions, one would look at intervals of $\pi$ associated with the tangent function; for this case, we state that the function does not exhibit standard periodic behavior.

Step 3

6.3 For what values of $x$ is $f(x) \cdot g(x) < 0$ for $x \in [-90^{\circ}; 120^{\circ}]?

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Answer

To analyze where $f(x) \cdot g(x) < 0$ , we need to evaluate the values of $x$ within the interval:

Determine intervals where $f(x) > 0$ and $g(x) < 0$ or vice versa.
Notably, $f(x)$ is positive for all $x$ since it approaches $90^{\circ}$ as $x$ increases, remaining above zero.
$g(x)$ transitions through zero, turning negative in the specified range.
The combined analysis reveals that $f(x) \cdot g(x) < 0$ occurs when $g(x)$ is negative alongside $f(x)$ being positive. This generally gives us ranges around critical points where $g(x) = 0$ .

Step 4

6.4 Write down the equation(s) of the asymptotes of $h \cdot h(x) = f(x + 10^{\circ})$ for $x \in [-90^{\circ}; 180^{\circ}]$

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Answer

For $h(x) = f(x + 10^{\circ})$ , the asymptotes would shift for the argument in $f$ . The vertical asymptotes of the function would occur where $f$ approaches its limits:

For the base function $f$ , asymptotes are at infinity.
Transitioning through the transformation $x + 10^{\circ}$ does not introduce new asymptotes but shifts the function leftward by $10^{\circ}$ .
Thus, the asymptote equation for the specific transformation would remain the same. Therefore, the general relationship leading to vertical behavior remains unchanged.

Given $f(x) = an^{-1} \left( \frac{1}{2} x \right)$ and $g(x) = \sin(x - 30^{\circ})$ for $x \in [-90^{\circ}; 180^{\circ}]$ 6.1 On the same set of axes draw the graphs of $f$ and $g$ - NSC Mathematics - Question 6 - 2017 - Paper 2

Worked Solution & Example Answer:Given $f(x) = an^{-1} \left( \frac{1}{2} x \right)$ and $g(x) = \sin(x - 30^{\circ})$ for $x \in [-90^{\circ}; 180^{\circ}]$ 6.1 On the same set of axes draw the graphs of $f$ and $g$ - NSC Mathematics - Question 6 - 2017 - Paper 2

6.1 On the same set of axes draw the graphs of $f$ and $g$

6.2 Write down the period of $f$

6.3 For what values of $x$ is $f(x) \cdot g(x) < 0$ for $x \in [-90^{\circ}; 120^{\circ}]?

6.4 Write down the equation(s) of the asymptotes of $h \cdot h(x) = f(x + 10^{\circ})$ for $x \in [-90^{\circ}; 180^{\circ}]$

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