## Analyzing the Function: y = tan⁻¹(4x)/(1+5x²) + tan⁻¹(2+3x)/(3-2x)

This article explores the properties of the function **y = tan⁻¹(4x)/(1+5x²) + tan⁻¹(2+3x)/(3-2x)**. We'll focus on understanding its domain, range, and key characteristics.

### Understanding the Components

The function is composed of two main parts, both involving the inverse tangent function (tan⁻¹):

**tan⁻¹(4x)/(1+5x²)**: This part represents the inverse tangent of a rational function. The denominator (1+5x²) ensures that the function is defined for all real values of x.**tan⁻¹(2+3x)/(3-2x)**: This part also involves the inverse tangent of a rational function. However, the denominator (3-2x) has a restriction. The function is undefined when 3-2x = 0, which occurs at x = 3/2.

### Domain of the Function

The domain of the function is the set of all real numbers where both components are defined. Since the first component is defined for all real values, the domain is restricted only by the second component.

Therefore, the **domain of the function is all real numbers except x = 3/2**. In interval notation: **(-∞, 3/2) U (3/2, ∞)**.

### Range of the Function

The range of the inverse tangent function is **(-π/2, π/2)**. Since we are adding two inverse tangent functions, the range of the combined function will also be within this interval.

However, the exact range is difficult to determine without further analysis. We can use calculus techniques, such as finding the derivative and analyzing its behavior, to determine the exact range.

### Key Characteristics

**Asymptotes:**Due to the presence of the inverse tangent functions, the function has horizontal asymptotes at y = -π/2 and y = π/2.**Critical Points:**To find critical points, we need to analyze the derivative of the function. The derivative will involve the chain rule and quotient rule for differentiation. Finding the critical points will help determine the intervals where the function is increasing or decreasing.**Inflection Points:**The second derivative of the function is used to identify inflection points, where the concavity of the graph changes.

### Further Analysis

For a deeper understanding, we can explore the following aspects:

**Graphing the function:**Using software or graphing calculators, we can visualize the function's behavior. This will provide insights into its increasing/decreasing intervals, concavity, and any local maximum or minimum points.**Limiting Behavior:**We can analyze the function's behavior as x approaches positive or negative infinity. This can help determine the presence of any horizontal or oblique asymptotes.**Applications:**Depending on the context, this function might have applications in various fields, such as physics, engineering, or economics.

Remember that understanding the domain, range, and key characteristics of a function is crucial for analyzing its behavior and applying it to real-world problems.