Use a graphing utility to graph the function and use the...

Use a graphing utility to graph the function and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function.
$$f(x)=-\frac{\left|x^{2}-9\right|}{\left|x^{2}+7\right|}$$

Key Concepts

Absolute Value Functions

Absolute value functions involve the absolute value operation, which outputs the non-negative magnitude of a number or expression. When included in functions, absolute values can create symmetry or piecewise behavior in the graph, often leading to distinct regions that need to be analyzed separately in terms of their properties.

Inverse Functions

An inverse function reverses the mapping of the original function, meaning that it 'undoes' the effect of the function. For a function to have an inverse that is also a function, it must be one-to-one. Inverse functions are fundamental in solving equations and understanding the symmetry of functions.

One-to-One Functions

A function is one-to-one (or injective) if each output value is produced by exactly one input value. This means that no horizontal line crosses the graph of the function more than once. One-to-one functions are important because only they have well-defined inverses that are also functions.

Graphing Functions

Graphing functions involves visualizing the relationship between dependent and independent variables by plotting points on the coordinate plane. It is useful for understanding a function's behavior, identifying key features such as intercepts, asymptotes, and intervals of increase or decrease, and for verifying algebraic properties through visual inspection.

Horizontal Line Test

The Horizontal Line Test is a graphical method used to determine if a function is one-to-one. If every horizontal line intersects the graph at most once, then the function is one-to-one and thus has an inverse function. This test provides a simple visual criterion to check injectivity.

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