Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Compare the graph of the function with the graph of \(f(x)=\arcsin x\) \(g(x)=-\arcsin x\)

Step-by-Step Solution

Answer

The function \(g(x)=-\arcsin x\) is the reflection of the function \(f(x)=\arcsin x\) over the x-axis. Both functions have the same domain, which is \([-1,1]\), and the same range, which is \([-π/2, π/2]\), but while \(f(x)=\arcsin x\) rises from \(-\pi/2\) to \(\pi/2\) as \(x\) goes from -1 to 1, \(g(x)=-\arcsin x\) falls from \(\pi/2\) to \(-\pi/2\) as \(x\) goes from -1 to 1.

Step 1: Understand the Graph of \(f(x)=\arcsin x\)

To graph the function \(f(x)=\arcsin x\), recall the shape of the graph of the arcsine function, defined for values of \(x\) in the interval \([-1,1]\) and ranging from \(-\pi/2\) to \(\pi/2\). The graph of the arcsine function is rising, with a slope that approaches infinity as \(x\) approaches either -1 or 1.

Step 2: Understand the Effect of the Negation

The function \(g(x) = -\arcsin x\) is a negation of \(f(x)=\arcsin x\). Negation mirrors the graph of \(f(x)=\arcsin x\) along the x-axis. This means, instead of rising from \(-\pi/2\) to \(\pi/2\) as \(x\) goes from -1 to 1, the graph of \(g(x) = -\arcsin x\) falls from \(\pi/2\) to \(-\pi/2\).

Step 3: Sketch the Graphs

Sketch the graph of \(f(x)=\arcsin x\), then sketch \(g(x)=-\arcsin x\), as described in step 2. Comparing these two graphs side by side will clearly show the effects of the negation on the original function. Note that the domain and range for both graphs are the same, but where the graph of \(f(x) = \arcsin x\) increases, the graph of \(g(x) = -\arcsin x\) decreases.

Key Concepts

Arcsine Function

The arcsine function, denoted as \( f(x) = \arcsin x \), is the inverse of the sine function, but only for a specific range. The arcsine function is defined only for values of \( x \) in the interval \([-1, 1]\). This is because the sine function produces outputs only within this interval when considering its inverse.

• Domain: The set of possible input values is \([-1, 1]\).
• Range: The set of possible output values is \([-\pi/2, \pi/2]\).
• The graph of \( \arcsin x \) starts at \(-\pi/2\) when \( x = -1 \) and rises continuously to \( \pi/2 \) as \( x = 1 \).
• The graph has a curved shape with a steep slope at the ends approaching infinity, meaning it becomes nearly vertical near \( x = -1 \) and \( x = 1 \).

These characteristics make the arcsine function an essential tool in trigonometry, especially when finding angles corresponding to sine values.

Function Negation

Negating a function involves multiplying it by \(-1\). This flips the graph of the function across the x-axis. For the arcsine function, the negated function is \( g(x) = -\arcsin x \). In simple terms, if you imagine the original graph of \( \arcsin x \), each point on the graph is mirrored downward across the x-axis.

• New Direction: Where \( \arcsin x \) originally increases from \(-\pi/2\) to \( \pi/2\), \(-\arcsin x \) now decreases from \( \pi/2\) to \(-\pi/2\).
• This transformation creates a reflection, offering a perfect mirror image that maintains the same domain and range.
• Negation affects the vertical symmetry of the function without altering the horizontal parameters; in this case, it produces a simple reversal of direction while keeping the graph within its established boundaries.
• This is evident in applications requiring phase shift or symmetry in oscillating functions.

Domain and Range

Understanding the domain and range is crucial when dealing with functions, especially those involving trigonometric inverses like the arcsine function. The domain specifies the values that an input \( x \) can take, while the range describes the possible outputs.For both \( f(x) = \arcsin x \) and \( g(x) = -\arcsin x \):

• Domain: The values \( x \) can assume are restrained to the closed interval \([-1, 1]\). This restriction stems from the fact that no sine value exists outside this interval, so the inverse must also work within these bounds.
• Range: The outputs extend from \(-\pi/2\) to \( \pi/2\) for \( f(x) = \arcsin x \), and conversely, from \( \pi/2\) to \(-\pi/2\) for \( g(x) = -\arcsin x \) due to the negation's effect. Despite this adjustment, the range envelope remains within \([-\pi/2, \pi/2]\).

Understanding these foundational aspects of domain and range helps in graphing and analyzing function behavior, ensuring accurate interpretations of direction and limitations in various mathematical contexts.