Understanding the Domain and Range of Composite Functions: A Comprehensive Tutorial

In the realm of mathematical analysis, understanding the domain and range of a function is vital for accurately determining its behavior and application. This tutorial delves into the intricacies of finding the domain and range of the composite function (f(g(x))), using the specific examples of (f(x) sqrt{x 1}) and (g(x) frac{1}{x}).

Introduction to the Problem

We aim to analyze the composite function (f(g(x))), where (f(x) sqrt{x 1}) and (g(x) frac{1}{x}). This requires a systematic approach to determine the valid input values (domain) and corresponding output values (range) for the composite function.

Step-by-Step Analysis

Step 1: Determine the Domain of (g(x))

The function (g(x) frac{1}{x}) is defined for all real numbers except (x 0). Therefore, the domain of (g(x)) is:

[ text{Domain of } g(x): x in mathbb{R} , x eq 0 ]

Step 2: Determine the Domain of (f(x))

The function (f(x) sqrt{x 1}) is defined when the expression inside the square root is non-negative, i.e., (x 1 geq 0). Solving this inequality, we get:

[ x geq -1 ]

Therefore, the domain of (f(x)) is:

[ text{Domain of } f(x): x in [-1, infty) ]

Step 3: Determine the Domain of (f(g(x)))

To find the domain of (f(g(x))), we need to ensure that the output of (g(x)) lies within the domain of (f(x)). Specifically, we need to find (x) such that (g(x) geq -1).

Let's solve this inequality step-by-step:

[ frac{1}{x} geq -1 ]

Multiplying both sides by (x) (keeping in mind the sign of (x)):

For (x > 0): [ 1 geq -x implies x geq -1 , text{(always true for } x > 0text{)} ] For (x

Combining these results, we get the valid intervals for (x): [ -infty

Therefore, the domain of (f(g(x))) is:

[ text{Domain of } f(g(x)): (-infty, -1) , cup , (0, infty) ]

Step 4: Determine the Range of (f(g(x)))

Next, we find the range of (f(g(x)) fleft(frac{1}{x}right) sqrt{frac{1}{x} 1}).

Let's analyze the behavior of the composite function:

For (x > 0): As (x) approaches (0^ ), (g(x) frac{1}{x} to infty), and thus (f(g(x)) to infty). As (x to infty), (g(x) to 0^ ), and thus (f(g(x)) to 1). For (x

Combining both intervals, the overall range of (f(g(x))) is:

[ text{Range of } f(g(x)): [0, 1] , cup , [1, infty) ]

Conclusion and Summary

In summary, we have:

[ text{Domain of } f(g(x)): (-infty, -1) , cup , (0, infty) ] [ text{Range of } f(g(x)): [0, 1] , cup , [1, infty) ]

This meticulous analysis ensures that we accurately determine the behavior of the composite function (f(g(x))) and understand its domain and range. Understanding these concepts is essential for advanced mathematical and computational applications.