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Magazine Chapter 6: Problem 50
Sketch the graph of the equation. $$y=\sin \left(\sin ^{-1} x\right)$$
Short Answer
Expert verified
The graph is a line segment of \( y = x \) from (-1, -1) to (1, 1).
Step by step solution
01
Understand the function
The equation given is \( y = \sin(\sin^{-1}(x)) \). Here, \( \sin^{-1}(x) \) is the inverse sine function (also known as arcsine, written as \( \arcsin(x) \)). This function takes an input \( x \) and returns the angle whose sine is \( x \). Then the sine function returns the sine of this angle, which is still \( x \). In simple terms, \( y = \sin(\sin^{-1}(x)) = x \).
02
Determine the domain
Since \( \sin^{-1}(x) \) is only defined for \( x \) in the range of \([-1, 1] \), the domain of \( y = \sin(\sin^{-1}(x)) \) is also \( x \in [-1, 1] \).
03
Analyze the range
For \( x \) within the domain \([-1, 1]\), the function \( y = x \) directly yields \( y \) values within the same range because \( x = y \). Thus, the range of the equation is \([ -1, 1 ]\).
04
Plot the graph
The equation simplifies to \( y = x \). Within the domain \([-1, 1]\), this is a straight line that starts at the point (-1, -1) and ends at the point (1, 1). To sketch the graph, plot these points and draw a straight line segment between them.
05
Review and label the graph
Ensure that the graph begins and ends at the points (-1, -1) and (1, 1), as these reflect the domain restrictions. Label the axes and mark the line segment to clearly indicate that the function is \( y = x \) over \( x \in [-1, 1] \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Arcsine Function
The arcsine function, denoted as \( \sin^{-1}(x) \) or \( \arcsin(x) \), is an important inverse trigonometric function. It plays a key role in reversing the sine function. When you use the arcsine function on a value \( x \), it provides the angle \( \theta \) for which the sine of \( \theta \) is exactly \( x \).
This means if \( \sin(\theta) = x \), then \( \theta = \sin^{-1}(x) \). However, not every number can be an input for the arcsine function. This leads us to its defined domain.
This means if \( \sin(\theta) = x \), then \( \theta = \sin^{-1}(x) \). However, not every number can be an input for the arcsine function. This leads us to its defined domain.
- The arcsine function only accepts values between -1 and 1, inclusive.
- The output, or the angle, will be between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \).
Domain and Range
Every mathematical function has a domain and a range, which are crucial for understanding its limits and behavior.
Let's break it down for \( y = \sin(\sin^{-1}(x)) \), which simplifies to \( y = x \):
Let's break it down for \( y = \sin(\sin^{-1}(x)) \), which simplifies to \( y = x \):
- Domain: The domain of a function explains what values \( x \) can take. For \( \sin^{-1}(x) \), \( x \) has a domain of \([-1, 1]\). This means the input \( x \) must be a value within this range.
- Range: The range describes the possible output values \( y \) of a function. Since \( y = x \) within the defined domain, the range is also \([-1, 1]\). Thus, any value of \( y \) you get will be between -1 and 1.
Graphing Functions
Graphing functions is a powerful way to visualize mathematical equations and understand their properties. For the equation \( y = \sin(\sin^{-1}(x)) \) or essentially \( y = x \), the graph is straightforward within the domain.
Here's how to graph it:
Here's how to graph it:
- Understand the line: Since the function simplifies to \( y = x \) over the interval \([-1, 1]\), it represents a linear relationship where every point on the line is a point whose x-coordinate is equal to its y-coordinate.
- Plotting points: Start by marking the critical points that form the edges of your domain: (-1, -1) and (1, 1). These points ensure that the line starts and ends at the domain's boundaries.
- Draw the graph: Connect these two points with a straight line. This line clearly shows the function's behavior over \([-1, 1]\). It's important to label your axes and mark these points to clearly convey that it is a segment and not an infinite line.
