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When examining a function like \( f(x) = \sqrt{1 - x^2} \), understanding the domain and range is crucial. The **domain** refers to all permissible values of \( x \) which make the function real and defined. Since the square root function \( \sqrt{u} \) is only defined when \( u \geq 0 \), we set: \( 1 - x^2 \geq 0 \). This simplifies to \( -1 \leq x \leq 1 \).
This interval means \( x \) can be any real number from -1 to 1, inclusive. The **range** includes all possible output values (i.e., \( f(x) \) values) you get by plugging the domain values into the function. Because the largest a square root function can get is when its argument is maximum, for \( \sqrt{1 - x^2} \), the highest value is 1 when \( x = 0 \). Thus, the range is \( 0 \leq f(x) \leq 1 \).
Understanding these concepts is fundamental when evaluating functions, as it ensures you're working within the boundaries that make the function valid.

Graphing functions is a powerful way to visualize relationships described by mathematical formulas. For \( f(x) = \sqrt{1 - x^2} \), graphing helps us see it as the top half of a circle.Here are some steps to graph this function:

• **Identify Key Points:** Check where the function starts and ends: the domain \( x = -1 \) and \( x = 1 \) both output \( y = 0 \). The middle point \( x = 0 \) gives \( y = 1 \).
• **Symmetry:** Note that the function is symmetric around the y-axis since it represents half of a circle. This means the graph on the left mirrors the right.
• **Plot the Curve:** The graph forms a smooth curve connecting the points \((-1,0)\), \((0,1)\), and \((1,0)\).

The symmetry and smooth nature of the upper semicurve make graphing simple once key points are plotted and connected.

The equation \( f(x) = \sqrt{1 - x^2} \) has its roots in circle equations. In general, a circle with a radius \( r \) centered at the origin has the equation \( x^2 + y^2 = r^2 \).For a unit circle (radius = 1), the equation becomes \( x^2 + y^2 = 1 \).

• **Solving for \( y \):** To get the equation for the top half of the circle, solve for \( y \): \( y = \sqrt{1 - x^2} \). Use only the positive square root for the top half, as \( y \) on the upper half is non-negative.
• **Circle Properties:** Circles are symmetric about both the x-axis and y-axis, reflecting the balanced nature of its points from the center.

The understanding of circle equations helps in recognizing and graphing the function as part of a geometric shape, making visual interpretation intuitive and enlightening.