(a) Use the horizontal line test to determine whether the...

(a) Use the horizontal line test to determine whether the function is one-to-one. (b) If the function is one-to-one, then graph the inverse of the function. (Remember that if $f$ is one-to-one and $(a, b)$ is on the graph of $f,$ then $(b, a)$ is on the graph of $f^{-1} .$) (Check your book for graph)

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Key Concepts

Horizontal Line Test

The horizontal line test is a method used to determine whether a function is one-to-one (injective). By drawing or imagining horizontal lines across the graph of a function, if any horizontal line intersects the graph at more than one point, the function is not one-to-one. This test is crucial because only one-to-one functions have inverses that are also functions.

Inverse Function

An inverse function reverses the roles of the inputs and outputs of the original function. If a function is one-to-one, then for every ordered pair (a, b) in the function, the inverse function will contain the ordered pair (b, a). Graphically, the inverse function is the reflection of the original function across the line y = x. This reflection demonstrates the swapping of the x and y coordinates effectively.

One-to-One Function

A one-to-one function, or injective function, is one where each output is produced by exactly one input. This means that if f(a) equals f(b) then a must equal b. The one-to-one property is necessary for a function to have an inverse that is also a function, ensuring that every element in the range corresponds to a unique element in the domain.

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