Question Statement

Draw the graph of each of the following functions against the intervals mentioned:

(i) $y={\cos }^{-1}\left(\frac{x+2}{2}\right)$ for $x\in [-2,0]$

(ii) $y={\sin }^{-1}\left(\frac{1}{x}\right)$ for $x\in [-2,2]$

(iii) $y={\tan }^{-1}\left(\frac{x^3}{2}\right)$ for $x\in \mathbb{R}$

(iv) $y={\cot }^{-1}\left(\frac{x+2}{3}\right)$ for $x\in \mathbb{R}$

(v) $y={\sec }^{-1}\left(\frac{3x-2}{2}\right)$ for $x\in \left(-\infty ,\frac{-2}{3}\right)\cup \left(\frac{2}{3},\infty \right)$

(vi) $y={\cos }^{-1}\left(\frac{4x+2}{5}\right)$ for $x\in \left[\frac{-7}{4},\frac{3}{4}\right]$

(vii) $y={\sin }^{-1}\left(\frac{x-1}{3}\right)$ for $x\in [-2,4]$

(viii) $y={\tan }^{-1}\left(3x+\frac{1}{2}\right)$ for $x\in \mathbb{R}$

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Background and Explanation

To graph inverse trigonometric functions, we identify the domain and range of the specific function. By substituting values of $x$ within the given interval, we calculate corresponding $y$ values to create a coordinate table, which is then plotted on a Cartesian plane.

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Solution

Part (i)

Function: $y={\cos }^{-1}\left(\frac{x+2}{2}\right)$ for $x\in [-2,0]$

To plot this, we calculate $y$ for various values of $x$ in the interval $[-2,0]$.

Part (ii)

Function: $y={\sin }^{-1}\left(\frac{1}{x}\right)$ for $x\in [-2,2]$

Note that for ${\sin }^{-1}(u)$, the domain is $|u|\le 1$. Here $u=1/x$, so $|1/x|\le 1$, which means $|x|\ge 1$. Therefore, values between $-1$ and $1$ (excluding $-1$ and $1$) are undefined.

Part (iii)

Function: $y={\tan }^{-1}\left(\frac{x^3}{2}\right)$ for $x\in \mathbb{R}$

We calculate points across a range of real numbers to see the cubic growth within the arctan function.

Part (iv)

Function: $y={\cot }^{-1}\left(\frac{x+2}{3}\right)$ for $x\in \mathbb{R}$

We can calculate the values by using the relationship $y={\tan }^{-1}\left(\frac{3}{x+2}\right)$.

Part (v)

Function: $y={\sec }^{-1}\left(\frac{3x-2}{2}\right)$ for $x\in \left(-\infty ,\frac{-2}{3}\right)\cup \left(\frac{2}{3},\infty \right)$

We can calculate the values by using the identity $y={\cos }^{-1}\left(\frac{2}{3x-2}\right)$.

Part (vi)

Function: $y={\cos }^{-1}\left(\frac{4x+2}{5}\right)$ for $x\in \left[\frac{-7}{4},\frac{3}{4}\right]$

Part (vii)

Function: $y={\sin }^{-1}\left(\frac{x-1}{3}\right)$ for $x\in [-2,4]$

Part (viii)

Function: $y={\tan }^{-1}\left(3x+\frac{1}{2}\right)$ for $x\in \mathbb{R}$

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Key Formulas or Methods Used

• Inverse Trigonometric Definitions:
  • ${\sin }^{-1}(x)$, ${\cos }^{-1}(x)$, ${\tan }^{-1}(x)$
• Reciprocal Identities:
  • ${\sec }^{-1}(x)={\cos }^{-1}\left(\frac{1}{x}\right)$
  • ${\csc }^{-1}(x)={\sin }^{-1}\left(\frac{1}{x}\right)$
  • ${\cot }^{-1}(x)={\tan }^{-1}\left(\frac{1}{x}\right)$ (Note: adjustments may be needed based on the quadrant/range).
• Point Plotting Method: Calculating specific $(x,y)$ coordinates to determine the shape and position of the curve.

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Summary of Steps

1. Identify the Interval: Determine the range of $x$-values provided for the function.
2. Check Domain Constraints: For functions like ${\sin }^{-1}$ and ${\cos }^{-1}$, ensure the inner expression stays within $[-1,1]$.
3. Calculate Coordinates: Substitute $x$ values into the function to find corresponding $y$ values.
4. Create a Table: Organize the calculated points for clarity.
5. Plot and Connect: Plot the points on a coordinate plane and connect them with a smooth curve consistent with the behavior of the specific inverse trigonometric function.