91Ó°ÊÓ

The **domain** of a function includes all possible inputs (x-values) that will give a real number output. Conversely, the **range** encompasses all possible outputs (y-values) from the function.
For the function \(f(x)=\frac{x+1}{x-2}\), the domain is restricted because division by zero is undefined. Therefore, \(f(x)\) is not defined at \(x = 2\), making the domain all real numbers except \(x = 2\).
The range of \(f(x)\) excludes \(y = 1\) because regardless of which x-value is used (except \(x=2\)), the function cannot output a value of 1. This is due to properties of the function's asymptote which will be discussed further.
Moving to its inverse \(f^{-1}(x)=\frac{2x+1}{x-1}\), similar logic is applied. Here, the domain includes all real numbers except \(x = 1\), as division by zero again is undefined.
Consequently, the range of \(f^{-1}(x)\) would exclude \(y = 2\). Understanding domains and ranges is crucial for graphing and identifying where functions might not exist.

Graphing a function involves plotting its possible outputs (y-values) against its inputs (x-values). This gives a visual representation of what the function does.
For the function \(f(x) = \frac{x+1}{x-2}\), a few key points like \((-1, -\frac{2}{3})\), \((0, -\frac{1}{2})\), and \((3, 4)\) can help sketch the curve. It is important to note the vertical asymptote at \(x = 2\), which indicates a point where the function is undefined. This is often shown as a dashed line on the graph.
When graphing \(f^{-1}(x)=\frac{2x+1}{x-1}\), points like \((-1, -3)\), \((0, -1)\), and \((2, 5)\) can guide the curve. It has its vertical asymptote at \(x=1\).
Visual graphs for both functions \(f(x)\) and \(f^{-1}(x)\) on the same axes provide insight into their behavior and range of values. Accurately plotting these functions aids in understanding their properties and any restrictions.

A key relationship between a function and its inverse is their symmetry concerning the line \(y = x\). This line acts as a mirror, reflecting the graph of a function into its inverse.
If you mentally trace the line \(y=x\) on a graph containing both \(f(x)\) and \(f^{-1}(x)\), you'll observe that each point \((a, b)\) on \(f(x)\) has a corresponding point \((b, a)\) on \(f^{-1}(x)\).
This property occurs because, in order for two functions to be inverses, swapping their x and y coordinates will yield the other function.
The symmetry is not just a visual cue but mathematically signifies that the composite functions \(f(f^{-1}(x))\) and \(f^{-1}(f(x))\) will both equal \(x\).
Observing these reflections on the graph can help confirm the correctness of an inverse, ensuring that it reflects exactly over \(y=x\). This understanding is fundamental when learning about functions and their inverses.