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In the context of functions, the **domain** refers to the set of all possible input values (or values of the independent variable) that will produce a valid output. For a function like \( f(x) = x \sqrt{1-x^2} \), the key consideration is the expression under the square root. The function is only defined when the square root has a non-negative value.
Therefore, we solve the inequality \(1-x^2 \geq 0 \). After rearranging, we see that \( x^2 \leq 1 \), which implies that \( x \) must lie within the interval \([-1, 1]\) for the function to be defined. This means our domain is \([-1, 1]\). Thus, understanding the domain is crucial because it tells us for which values of \(x\) the function exists.

Asymptotes are lines that a graph approaches but never actually reaches. They often appear in functions involving fractions or logarithms. There are two common types:

• **Vertical Asymptotes**: These occur when the function approaches infinity or negative infinity as \( x \) nears a specific value.
• **Horizontal Asymptotes**: These appear when \( y \) approaches a constant value as \( x \) goes towards infinity or negative infinity.

In our function \( f(x) = x \sqrt{1-x^2} \), there are no asymptotes because the function is confined to \([-1, 1]\) and defined by the boundary of this interval. Therefore, it doesn't extend towards infinity or negative infinity within this domain.

A **sign diagram** helps us visualize where a function is positive, negative, or zero. It's like a roadmap of the function's behavior within its domain.
In our function, \( f(x) = x \sqrt{1-x^2} \), we find:

• The function is zero at \( x = -1, 0, \text{ and } 1 \).
• For \( -1 < x < 0 \), the function is negative because \( x \) is negative and the square root is positive, making the product negative.
• For \( 0 < x < 1 \), \( x \) and the square root are both positive, so the product is positive.

Using this information, a sign diagram allows us to see the intervals of positivity and negativity, providing useful insight when sketching the graph.

When creating a graph of a function, we integrate understanding from the domain, sign diagram, and potential asymptotes. With \( f(x) = x \sqrt{1-x^2} \), here’s how the graph takes shape:

• The domain \([-1, 1]\) confines the graph to this interval on the x-axis.
• There are important intercepts at \( x = -1, 0, \text{ and } 1 \), all touching the x-axis.
• The lack of asymptotes ensures the function smoothly transitions without infinitely approaching any specific line.
• The shape of the graph mimics a semi-ellipse due to the structure of the square root, concave at the origin.

Grasping these characteristics allows for an accurate sketch, visualizing how \( f(x) \) behaves within its defined range.