Domain and Range of the Function f(x) √[x/(x-2)]

Understanding the behavior of mathematical functions is crucial in various fields, including calculus, real analysis, and optimization. This article will delve into the domain and range of the function f(x) √[x/(x-2)].

Domain of the Function

First, we need to determine the domain of the function. The domain of a function consists of all possible values of x for which the function is defined. To find the domain of the given function, we need to ensure that the expression inside the square root is non-negative and that the denominator is not zero.

Step 1: Analyze the Expression Inside the Square Root

We start with the expression inside the square root:

frac{x}{x-2} ≥ 0

To determine when this fraction is non-negative, we need to find the values of x that satisfy this inequality.

Step 2: Solve the Inequality frac{x}{x-2} ≥ 0

First, we identify the critical points where the expression changes sign. These points are where the numerator or the denominator is zero:

x 0 x - 2 0 Rightarrow x 2

These points split the real number line into three intervals: (-infty, 0), (0, 2), and (2, infty). We need to test the sign of the expression in each of these intervals.

Interval (-infty, 0): Choose x -1: frac{-1}{-1-2} frac{-1}{-3} frac{1}{3} 0

Interval (0, 2): Choose x 1: frac{1}{1-2} frac{1}{-1} -1 0

Interval (2, infty): Choose x 3: frac{3}{3-2} frac{3}{1} 3 0

Conclusion

The expression frac{x}{x-2} is non-negative in the intervals (-infty, 0) and (2, infty). Therefore, the domain of f(x) is:

Domain: (-infty, 0) cup (2, infty)

We also need to ensure that the denominator is not zero:

x - 2 ≠ 0 Rightarrow x ≠ 2

Combining these conditions, the final domain is:

Domain of f(x) √[x/(x-2)]: (-infty, 0) cup (2, infty)

Range of the Function

To determine the range of the function, we need to analyze the possible values of f(x).

Step 1: Analyzing the Square Root Behavior

Let's rewrite the function in a more convenient form:

f(x) √[x/(x-2)]

For some values of x in the domain:

Case 1: For x 0

Let x -2tan^2θ where θ in [0, frac{π}{2}]:

f(x) √[frac{-2 tan^2 θ}{2(tan^2 θ - 1)}] √[sin^2 θ] sin θ in [0, 1]

Case 2: For x 2

Let x 2sec^2θ where θ in [0, frac{π}{2}]:

f(x) √[frac{2 sec^2 θ}{2(sec^2 θ - 1)}] √[frac{1}{sin^2 θ}] frac{1}{sin θ} in (1, ∞)

Conclusion

From the above analysis, we can conclude that the range of the function f(x) is:

Range: [0, 1] cup (1, ∞)

Summary

After a thorough analysis, we have determined that the domain and range of the function f(x) √[x/(x-2)] are as follows:

Domain: (-infty, 0) cup (2, infty) Range: [0, 1] cup (1, ∞)

These results provide a comprehensive understanding of the function's behavior and its limitations.