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Function analysis involves examining the characteristics and behavior of a mathematical function. Understanding whether functions are one-to-one is an essential part of this analysis.
A function is labeled as "one-to-one" if every input maps to a unique output. Conversely, a function is not one-to-one if two or more different inputs can result in the same output.
By determining if a function is one-to-one, we can better understand its structure and behavior. In the example given, the function \( f(x) = \sqrt{36-x^2} \) is analyzed. This function arises from the equation of a circle and represents part of its graph. Since different inputs, such as 3 and -3, produce the same output, the function is not one-to-one. This violates the criteria for one-to-one functions.

The domain and range of a function are fundamental concepts that define its inputs and outputs respectively.
The domain refers to all possible input values that can be used in a function. For the function \( f(x) = \sqrt{36-x^2} \), the domain includes values of \( x \) satisfying \( -6 \leq x \leq 6 \). This restriction ensures that the expression under the square root is non-negative,
which is a requirement for real-valued square roots.

• Domain: The set of all \( x \)-values which make the function defined.

The range indicates all possible output values of the function. Here, since the function describes the top half of a circle with radius 6, the range is \( 0 \leq f(x) \leq 6 \), because square root values will be non-negative.

• Range: The set of potential resulting \( y \)-values after substituting the domain into the function.

The equation \( x^2 + y^2 = r^2 \) represents a circle with radius \( r \) centered at the origin. This fundamental equation helps to visually connect the function to geometry.
For the given function \( f(x) = \sqrt{36-x^2} \), we relate it to this circle equation as \( x^2 + y^2 = 36 \).

• The term \( 36 \) indicates the radius squared, hence radius \( r = 6 \).

In this specific function, \( y = \sqrt{36-x^2} \) corresponds to the upper half of a circle. This is because the square root selects positive values of \( y \), indicating points above the x-axis.

• Equation element: \( x^2 + y^2 = 36 \) describes a full circle.
• Function representation: \( \sqrt{36-x^2} \) captures just the top semi-circle.

This connection is crucial for understanding how functions can graphically represent more complex shapes, like circles.