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A function is considered a "one-to-one" function if every unique input correlates to a unique output. This means if we have two different inputs, say \(a\) and \(b\), the function values \(f(a)\) and \(f(b)\) must also be different as long as \(a eq b\).
This characteristic is crucial when determining the inverse of a function.

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To check if a function is one-to-one, we can use the horizontal line test.

• If any horizontal line crosses the graph of the function more than once, the function is not one-to-one.

When a function is one-to-one, each output corresponds to exactly one input, which ensures that its inverse will be a proper function.
In other words, one-to-one functions and their inverses can swap the roles of inputs and outputs perfectly.

Square root functions are special kinds of functions represented by the equation \(f(x) = \sqrt{x}\). The defining feature is the square root symbol \(\sqrt{}\), which means we are looking for numbers that multiply by themselves to give the number under the root.
This function has a distinctive shape, with the graph only existing for non-negative values of \(x\).

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• The square root function starts at the origin (0,0) and continues to infinity in the positive x-direction.
• The values on the y-axis increase as x becomes larger, but the increase happens at a decreasing rate.

Understanding this function helps in finding its inverse because:

• The inverse reverses the operation. In the case of square roots, the inverse is squaring the numbers.

Therefore, knowing how square root functions behave guides us to properly invert its inputs and outputs.

In mathematics, understanding the domain and range of a function is crucial. The domain represents all possible input values (x-values) that a function can accept, while the range represents all possible output values (y-values).
For the function \(f(x) = \sqrt{x}\), the domain is all non-negative numbers, \(x \geq 0\), because you cannot take the square root of a negative number without venturing into complex numbers.
Likewise, the range is also all non-negative numbers because the square root of any non-negative number cannot be negative.

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When determining the domain and range of an inverse function such as \(f^{-1}(x) = x^2\), we perform a swap:

• The domain of the original function becomes the range of the inverse.
• The range of the original function becomes the domain of the inverse.

For \(f^{-1}(x) = x^2\), this means the domain is \(x \geq 0\), and the range is \(y \geq 0\), maintaining the structure needed for these connected functions.