Exploring the Domain and Range of (y frac{x}{x^2 - x - 1}) | A Comprehensive Guide

Understanding the domain and range of a function is crucial for grasping its behavior and significance in mathematics. In this article, we will explore the function (y frac{x}{x^2 - x - 1}) in detail, examining its domain and range step by step.

Domain Analysis

The domain of a function consists of all the values of (x) for which the function is defined. For the function (y frac{x}{x^2 - x - 1}), the only restriction is the denominator, which cannot be zero.

To find when the denominator is zero, we solve:

(x^2 - x - 1 0)

The discriminant of the quadratic equation is given by:

(D b^2 - 4ac (-1)^2 - 4 cdot 1 cdot (-1) 1 4 5)

Since the discriminant is positive ((D > 0)), the quadratic has two real roots. However, we need to find these roots explicitly:

(x frac{-b pm sqrt{D}}{2a} frac{1 pm sqrt{5}}{2})

The quadratic factorizes to:

(x^2 - x - 1 (x - frac{1 sqrt{5}}{2})(x - frac{1 - sqrt{5}}{2}))

Thus, the function is undefined at:

(x frac{1 sqrt{5}}{2}) and (x frac{1 - sqrt{5}}{2})

The domain is all real numbers except these two points:

(text{Domain: } (-infty, frac{1 - sqrt{5}}{2}) cup (frac{1 - sqrt{5}}{2}, frac{1 sqrt{5}}{2}) cup (frac{1 sqrt{5}}{2}, infty))

Range Analysis

The range of a function is the set of all possible output values. To determine the range of (y frac{x}{x^2 - x - 1}), we analyze its behavior as (x rightarrow pm infty), identify critical points, and evaluate the function at these points.

First, as (x rightarrow pm infty), the function behaves like:

(y approx frac{x}{x^2} frac{1}{x} rightarrow 0)

This suggests that (y) approaches 0 as (x) goes to positive or negative infinity.

Next, we differentiate (y) to find any critical points:

(y' frac{x^2 - x - 1 - 2x^2 x}{(x^2 - x - 1)^2} frac{-x^2 - 1}{(x^2 - x - 1)^2})

Solving for critical points:

( -x^2 - 1 0 Rightarrow x pm sqrt{-1} Rightarrow text{No real roots})

We still need to check the behavior of the function at the points where the denominator is zero:

( x frac{1 pm sqrt{5}}{2} )

Evaluating the function at these points:

(x frac{1 sqrt{5}}{2}) (y frac{frac{1 sqrt{5}}{2}}{(frac{1 sqrt{5}}{2})^2 - (frac{1 sqrt{5}}{2}) - 1} 1) (x frac{1 - sqrt{5}}{2}) (y frac{frac{1 - sqrt{5}}{2}}{(frac{1 - sqrt{5}}{2})^2 - (frac{1 - sqrt{5}}{2}) - 1} -frac{1}{3})

By analyzing the behavior of the function in the intervals between these critical points and as (x rightarrow pm infty), we conclude that the range is:

(text{Range: } (-frac{1}{3}, 1))

Summary

The domain and range of the function (y frac{x}{x^2 - x - 1}) are:

(text{Domain: } (-infty, frac{1 - sqrt{5}}{2}) cup (frac{1 - sqrt{5}}{2}, frac{1 sqrt{5}}{2}) cup (frac{1 sqrt{5}}{2}, infty)) (text{Range: } (-frac{1}{3}, 1))