Given: $f(x) = \tan x$ and $g(x) = \cos(x - 45^\circ)$ for $x \in [0^\circ; 360^\circ]$ 5.1 Draw sketch graphs of $f$ and $g$ on the same set of axes on the grid provided in the ANSWER BOOK. Clearly indicate ALL turning points, end points, asymptotes and intercepts with the axes. 5.2 Use your graphs to write down the value(s) of $x$ for which: 5.2.1 $f$ is undefined 5.2.2 $f(x)g(x) \leq 0$ where $x \in [90^\circ; 180^\circ]$

NSC Technical Mathematics Functions and Graphs: Given: $f(x) = \tan x$ and $g(

Given: $f(x) = \tan x$ and $g(x) = \cos(x - 45^\circ)$ for $x \in [0^\circ; 360^\circ]$ 5.1 Draw sketch graphs of $f$ and $g$ on the same set of axes on the grid provided in the ANSWER BOOK - NSC Technical Mathematics - Question 5 - 2021 - Paper 2

Question 5

$Given:---$f(x)-=-\tan-x$-and-$g(x)-=-\cos(x---45^\circ)$-for-$x-\in-[0^\circ;-360^\circ]$----5.1-Draw-sketch-graphs-of-$f$-and-$g$-on-the-same-set-of-axes-on-the-grid-provided-in-the-ANSWER-BOOK-NSC Technical Mathematics-Question 5-2021-Paper 2.png$

Given: $f(x) = \tan x$ and $g(x) = \cos(x - 45^\circ)$ for $x \in [0^\circ; 360^\circ]$ 5.1 Draw sketch graphs of $f$ and $g$ on the same set of axes on the gri... show full transcript

Worked Solution & Example Answer:Given: $f(x) = \tan x$ and $g(x) = \cos(x - 45^\circ)$ for $x \in [0^\circ; 360^\circ]$ 5.1 Draw sketch graphs of $f$ and $g$ on the same set of axes on the grid provided in the ANSWER BOOK - NSC Technical Mathematics - Question 5 - 2021 - Paper 2

Step 1

5.1 Draw sketch graphs of $f$ and $g$

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Answer

To sketch $f(x) = \tan x$ , note that the function has vertical asymptotes at $x = 90^\circ$ and $x = 270^\circ$ . The function is periodic with a shape that approaches infinity near the asymptotes. The x-intercepts occur at $x = 0^\circ$ , $x = 180^\circ$ , etc.

For $g(x) = \cos(x - 45^\circ)$ , the function has its x-intercepts at $x = 135^\circ$ and $x = 315^\circ$ and ranges from -1 to 1. It has no asymptotes. Indicate the turning points at $x = 45^\circ$ and $x = 225^\circ$ along with the respective endpoints.

Step 2

5.2.1 $f$ is undefined

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Answer

The function $f(x) = \tan x$ is undefined at the vertical asymptotes, which occur at $x = 90^\circ$ and $x = 270^\circ$ .

Step 3

5.2.2 $f(x)g(x) \leq 0$ where $x \in [90^\circ; 180^\circ]$

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Within the interval $[90^\circ; 180^\circ]$ , $f(x)$ is negative at $90^\circ$ and becomes zero at $180^\circ$ . The values of $x$ satisfying $f(x)g(x) \leq 0$ include the interval $90^\circ < x < 135^\circ$ or $x = 180^\circ$ .

Given: $f(x) = \tan x$ and $g(x) = \cos(x - 45^\circ)$ for $x \in [0^\circ; 360^\circ]$ 5.1 Draw sketch graphs of $f$ and $g$ on the same set of axes on the grid provided in the ANSWER BOOK - NSC Technical Mathematics - Question 5 - 2021 - Paper 2

Worked Solution & Example Answer:Given: $f(x) = \tan x$ and $g(x) = \cos(x - 45^\circ)$ for $x \in [0^\circ; 360^\circ]$ 5.1 Draw sketch graphs of $f$ and $g$ on the same set of axes on the grid provided in the ANSWER BOOK - NSC Technical Mathematics - Question 5 - 2021 - Paper 2

5.1 Draw sketch graphs of $f$ and $g$

5.2.1 $f$ is undefined

5.2.2 $f(x)g(x) \leq 0$ where $x \in [90^\circ; 180^\circ]$

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