Problem 44 - Precalculus with Calculus Previews

Question

Sketch the graph of the given function. $$ y=\sin ^{-1}(x+1) $$

Step-by-Step Solution

Answer

The graph of \( y=\sin^{-1}(x+1) \) is defined from \( x=-2 \) to \( x=0 \) and ranges from \( y=-\frac{\pi}{2} \) to \( y=\frac{\pi}{2} \). It is an increasing curve.

Step 1: Understand the function

The function given is \( y = \sin^{-1}(x+1) \). This function is an inverse sine function, known as arcsin. The arcsin function returns the angle whose sine is a given number. Here, the input to the arcsin function is \( x+1 \).

Step 2: Determine the domain of the function

Since the sine function has a range of \([-1, 1]\), the input to \( \sin^{-1} \), which is \( x+1 \), must lie between \(-1\) and \( 1\) inclusive. Therefore, the domain of \( y = \sin^{-1}(x+1) \) is the set of \( x \) values for which \(-1 \leq x+1 \leq 1\). Solving these inequalities, we get \(-2 \leq x \leq 0\).

Step 3: Determine the range of the function

The range of \( y = \sin^{-1}(x+1) \) is given by the range of the \( \sin^{-1} \) function, which is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This is because \( \sin^{-1} \) outputs angles in the principal range from \(-\frac{\pi}{2}\) to \( \frac{\pi}{2}\).

Step 4: Identify key points

To sketch the graph, identify key points within the domain. Evaluate the function at the endpoints of the domain: - At \( x = -2 \), \( y = \sin^{-1}((-2)+1) = \sin^{-1}(-1) = -\frac{\pi}{2} \).- At \( x = 0 \), \( y = \sin^{-1}(0+1) = \sin^{-1}(1) = \frac{\pi}{2} \).- At \( x = -1 \), \( y = \sin^{-1}(-1+1) = \sin^{-1}(0) = 0 \).These points help shape the curve.

Step 5: Sketch the graph

Using the calculated domain \([-2, 0]\) and range \([-\frac{\pi}{2}, \frac{\pi}{2}]\), plot the function using the key points: - Plot \((-2, -\frac{\pi}{2})\), \((-1, 0)\), and \((0, \frac{\pi}{2})\). - The graph will start at \((-2, -\frac{\pi}{2})\), go through \((-1, 0)\), and end at \((0, \frac{\pi}{2})\), showing an increasing trend.

Key Concepts

Graph Sketching

Sketching the graph of the inverse sine function, also known as arcsin, requires understanding both the inputs and outputs of the function. For the function \( y = \sin^{-1}(x+1) \), graph sketching involves knowing the relationship between \( x \) and \( y \). First, identify key points by evaluating the function at specific values of \( x \) within its domain. This will give significant points that the graph passes through.

• At \( x = -2 \), the point is \((-2, -\frac{\pi}{2})\).
• At \( x = -1 \), the point is \((-1, 0)\).
• At \( x = 0 \), the point is \((0, \frac{\pi}{2})\).

Using these points, sketch the curve of the function, ensuring it gradually rises, showing a clear increase from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). The graph will start low and end high, reflecting the positive slope of the arcsin function within its domain.

Function Domain

The domain of the inverse sine function is determined by the restrictions on its input values. For the function \( y = \sin^{-1}(x+1) \), the expression inside the arcsin, \( x+1 \), must be restricted to ensure validity. The inverse sine function, arcsin, can only accept inputs in the range \([-1, 1]\).
To find the domain of \( y = \sin^{-1}(x+1) \):

• Start by setting up the inequality: \(-1 \leq x + 1 \leq 1\).
• Solve for \( x \) to find \(-2 \leq x \leq 0\).

This means the function is only defined for \( x \) values between \(-2\) and \(0\), inclusive. Understanding the domain helps us know where the function is applicable in any real-world context.

Range of Function

The range of the arcsin function specifies the possible outputs or values of \( y \). For any arcsin function, \( \sin^{-1}(u) \), the output angle \( y = \sin^{-1}(u) \) is constrained to the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
This range is vital as it provides the principal values for which arcsin is defined. Applied to the function \( y = \sin^{-1}(x+1) \):

• The range is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), meaning the output angles of the function can only reside within these angles.

Remember, this range restriction ensures the function maps to real and physically meaningful angles when graphed or calculating values.

Arcsin Function

The arcsin function is the inverse of the sine function, meaning it helps us find the angle whose sine value equals the given input. Signified by \( \sin^{-1}(x) \), the arcsin function plays an essential role in trigonometry by converting a sine value back to an angle.
Key characteristics of the arcsin function include:

• Its domain: \([-1, 1]\) - the valid range of sine values for which an angle can be found.
• Its range: \([-\frac{\pi}{2}, \frac{\pi}{2}]\) - the angle results that fall between these values.

Applying this to \( y = \sin^{-1}(x+1) \), complexities arise from altering the input \( x \), shifting it via \( x+1 \). Understanding the arcsin's properties digests such transformations, ensuring accurate graphing and analytical evaluations.