Let f(x)=(x)/(1-x^2). (a) From a graph of f, explain why f...

Let $f(x)=\frac{x}{1-x^{2}}$. (a) From a graph of $f,$ explain why $f$ is not invertible. Furthermore, explain why $f$ is invertible if we restrict the domain to (-1,1) . (b) Let $h$ be the inverse function for $f$ restricted to $(-1,1) .$ What are $h(-10), h(-2), h(0), h(3), h(9) ?$ (c) Use the quadratic formula to find an expression for $h(x)$.

Let $f(x)=\frac{x}{1-x^{2}}$.
(a) From a graph of $f,$ explain why $f$ is not invertible. Furthermore, explain why $f$ is invertible if we restrict the domain to (-1,1) .
(b) Let $h$ be the inverse function for $f$ restricted to $(-1,1) .$ What are $h(-10), h(-2), h(0), h(3), h(9) ?$
(c) Use the quadratic formula to find an expression for $h(x)$.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations of the form ax² + bx + c = 0. When finding an inverse function, one might arrive at a quadratic equation after interchanging the roles of x and y. The quadratic formula, x = (-b ± ?(b² - 4ac)) / (2a), provides a method to obtain the explicit expression for the inverse, with the correct root chosen according to the domain restrictions.

Inverse Function Concept

The inverse function of a function f is another function, usually denoted f?¹, that reverses the action of f. This means that if f maps an input x to an output y, then f?¹ maps y back to x. Conceptually, finding the inverse requires solving the equation y = f(x) for x in terms of y, while taking into account the correct domain and range.

Invertibility

A function is invertible if and only if it is one-to-one, which means that each output is produced by exactly one input. This ensures that the inverse function, which reverses the original function's effect, is well-defined. When a function fails to be one-to-one, it means that multiple inputs map to the same output, making it impossible to uniquely determine the input from an output.

Horizontal Line Test

The horizontal line test is a graphical method used to determine if a function is one-to-one, and therefore invertible. If any horizontal line intersects the graph of the function more than once, the function is not one-to-one. Conversely, if every horizontal line intersects the graph at most once, the function passes the test and is invertible.

Domain Restriction

Restricting the domain of a function can force it to become one-to-one when it is not one-to-one over its entire domain. By limiting the inputs, you eliminate the cases where different inputs produce the same output. In the context of inversion, careful domain restriction ensures that the inverse function exists and is well-defined.

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