(ii) F(x)=sqrt((9-3^(x))/(5^(-x)-125))

6 min read Jun 16, 2024

Analyzing the Function f(x) = sqrt((9-3^(x))/(5^(-x)-125))

This article will explore the function f(x) = sqrt((9-3^(x))/(5^(-x)-125)) by examining its domain, range, key features, and potential graphs.

1. Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. To determine the domain of f(x), we need to consider the following restrictions:

The radicand (expression under the square root) must be non-negative: This means (9 - 3^(x)) / (5^(-x) - 125) ≥ 0.
The denominator cannot be zero: This means 5^(-x) - 125 ≠ 0.

Let's break down each restriction:

Radicand Non-Negative:
- Case 1: Numerator ≥ 0: 9 - 3^(x) ≥ 0 => 3^(x) ≤ 9 => x ≤ 2 (since 3^2 = 9)
- Case 2: Denominator > 0: 5^(-x) - 125 > 0 => 5^(-x) > 125 => -x > log₅(125) => x < -3 (since log₅(125) = 3)
- Case 3: Numerator ≤ 0 and Denominator ≤ 0: This case would make the entire expression non-negative. We need to consider when 9 - 3^(x) ≤ 0 and 5^(-x) - 125 ≤ 0. This results in x ≥ 2 and x ≤ -3, which is not possible.
Denominator Non-Zero: 5^(-x) - 125 ≠ 0 => 5^(-x) ≠ 125 => -x ≠ log₅(125) => x ≠ -3

Combining all the restrictions, the domain of f(x) is: x ∈ (-∞, -3) ∪ (-3, 2]

2. Range

The range of a function is the set of all possible output values (y-values) that the function can produce. To determine the range of f(x), we need to consider the following:

The square root function always produces non-negative values.
The function is defined for all x within its domain, and it can take on any non-negative value as a result of the square root.

Therefore, the range of f(x) is: y ∈ [0, ∞)

3. Key Features

Asymptotes: The function has a vertical asymptote at x = -3 due to the denominator approaching zero as x approaches -3.
Intercepts: To find the y-intercept, we set x = 0 and get f(0) = sqrt((9-1)/(1-125)) = sqrt(-8/124), which is undefined. Therefore, the function has no y-intercept. To find the x-intercept, we set f(x) = 0 and solve for x. This gives us 9 - 3^(x) = 0 => x = 2. Therefore, the function has an x-intercept at x = 2.
Monotonicity: The function is monotonically decreasing on its domain. This can be confirmed by analyzing the derivative of the function.

4. Graphing the Function

To graph the function, we can use the domain, range, key features, and some additional points. The graph of the function will look like this:

The graph will be above the x-axis because the range is non-negative.
The graph will have a vertical asymptote at x = -3.
The graph will pass through the x-intercept at x = 2.
The graph will be decreasing on its domain.

Note: The exact shape of the graph requires further analysis of the function's behavior near the asymptote and the x-intercept.

In conclusion, the function f(x) = sqrt((9-3^(x))/(5^(-x)-125)) is a complex function with a limited domain and a non-negative range. It exhibits a vertical asymptote, an x-intercept, and decreasing behavior on its domain. By understanding its key features, we can sketch a general representation of the function's graph.