Question Statement

Find the domain and range of the following functions graphically:

(i) $f(x)=\sin \left(\frac{x}{2}\right)$

(ii) $g(x)=3\cos \left(\frac{x}{3}\right)$

(iii) $h(x)=2\tan x$

(iv) $y=\cot \left(\frac{x}{4}\right)$

(v) $y=2\sec (2x)$ for the interval $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

(vi) $y=\sin 2x$ for the interval $(-\pi ,\pi )$

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Background and Explanation

To find the domain and range of trigonometric functions graphically, we plot specific points $(x,y)$ by substituting values into the function. The domain is the set of all possible input values ($x$) for which the function is defined, and the range is the set of all resulting output values ($y$), representing the vertical extent of the graph.

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Solution

Part (i) $f(x)=\sin \left(\frac{x}{2}\right)$

To graph this function, we calculate values for $f(x)$ over the interval $[-2\pi ,2\pi ]$:

Based on the graph and the nature of the sine function:

• Domain $=\mathbb{R}$
• Range $=[-1,1]$

Part (ii) $g(x)=3\cos \frac{x}{3}$

We calculate the values for $g(x)$ over a wider interval to observe the period:

$\begin{array}{rl}& \text{ Domain }=\mathbb{R}\\ & \text{ Range }=[-3,3]\end{array}$

Part (iii) $h(x)=2\tan x$

The tangent function has vertical asymptotes where $\cos x=0$.

$\begin{array}{rl}& \text{ Domain }=\mathbb{R}-\left\{(2n+1)\frac{\pi }{2}\mid n\in \mathbb{Z}\right\}\\ & \text{ Range }=\mathbb{R}\end{array}$

Part (iv) $y=\cot \left(\frac{x}{4}\right)$

The cotangent function has vertical asymptotes where $\sin (\frac{x}{4})=0$.

• Domain $=\mathbb{R}-\{4n\pi \mid n\in \mathbb{Z}\}$
• Range $=\mathbb{R}$

Part (v) $y=2\sec (2x)$

This function is evaluated on the interval $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$.

$\begin{array}{rl}& \text{ Domain }y=\mathbb{R}-\left\{\frac{(2n+1)}{4}\pi \mid n\in \mathbb{Z}\right\}\\ & \text{ Range }y=(-\infty ,\infty )\end{array}$

Part (vi) $y=\sin 2x$

This function is evaluated on the interval $(-\pi ,\pi )$.

• Domain $y=\mathbb{R}$
• Range $y=[-1,1]$ (Note: Raw data suggests $\mathbb{R}$, but the graph and table confirm the standard sine range).

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Key Formulas or Methods Used

• Sine/Cosine Range: For $y=A\sin (Bx)$ or $y=A\cos (Bx)$, the range is $[-A,A]$.
• Tangent Domain: $y=\tan (x)$ is undefined at $x=\frac{\pi }{2}+n\pi$.
• Secant Domain: $y=\sec (x)$ is undefined where $\cos (x)=0$.
• Cotangent Domain: $y=\cot (x)$ is undefined where $\sin (x)=0$.
• Point Plotting: Creating a table of values to visualize the function's behavior.

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Summary of Steps

1. Select $x$-values: Choose a variety of input values, including those that might result in asymptotes (where the denominator of the trig function is zero).
2. Calculate $y$-values: Substitute $x$ into the function to find the corresponding $y$ coordinates.
3. Plot the Graph: Draw the points on a coordinate system and connect them to show the function's shape.
4. Identify Domain: Determine all $x$-values where the graph exists.
5. Identify Range: Determine the interval of $y$-values covered by the graph from its lowest to highest points.