Continuity And Differentiability Analysis Of Composite Functions

$$f(x) = \begin{cases} x+2, & 0 < x < 2 \\ 6-x, & x \ge 2 \end{cases}$$

$$g(x) = \begin{cases} 1+ \tan x, & 0 \le x < \frac{\pi}{4} \\ 3 - \cot x, & \frac{\pi}{4} \le x < \pi \end{cases}$$

This article delves into the properties of the composite function f(g(x)), specifically focusing on its continuity and differentiability within the interval [0, π]. We will analyze the behavior of f(g(x)) by examining the individual functions f(x) and g(x), and then explore their composition. Understanding the characteristics of composite functions is crucial in various fields of mathematics, including calculus and real analysis. This analysis will provide insights into the conditions under which composite functions maintain the properties of continuity and differentiability.

Analyzing the Continuity of f(g(x))

When we analyze the continuity of the composite function f(g(x)), we must consider the continuity of both f(x) and g(x) individually, as well as the range of g(x) within the domain of f(x). The function f(x) is defined piecewise, with f(x) = x + 2 for 0 < x < 2 and f(x) = 6 - x for x ≥ 2. It is continuous in its respective intervals since both x + 2 and 6 - x are linear functions and hence continuous. However, we need to check the continuity at the point x = 2. The left-hand limit at x = 2 is lim (x→2-) f(x) = 2 + 2 = 4, and the right-hand limit at x = 2 is lim (x→2+) f(x) = 6 - 2 = 4. Also, f(2) = 6 - 2 = 4. Since the left-hand limit, the right-hand limit, and the function value at x = 2 are all equal, f(x) is continuous at x = 2 and therefore continuous in its entire domain.

Now, let's analyze g(x). It is defined piecewise as well, with g(x) = 1 + tan x for 0 ≤ x < π/4 and g(x) = 3 - cot x for π/4 ≤ x < π. The function 1 + tan x is continuous in the interval [0, π/4) since tan x is continuous in this interval. Similarly, 3 - cot x is continuous in the interval (π/4, π) since cot x is continuous there. We need to check the continuity of g(x) at x = π/4. The left-hand limit at x = π/4 is lim (x→(π/4)-) g(x) = 1 + tan(π/4) = 1 + 1 = 2. The right-hand limit at x = π/4 is lim (x→(π/4)+) g(x) = 3 - cot(π/4) = 3 - 1 = 2. Also, g(π/4) = 3 - cot(π/4) = 2. Since the left-hand limit, the right-hand limit, and the function value at x = π/4 are all equal, g(x) is continuous at x = π/4 and thus continuous in its domain [0, π).

To analyze the continuity of f(g(x)), we need to determine the range of g(x). For 0 ≤ x < π/4, g(x) = 1 + tan x. Since tan x varies from 0 to 1 in this interval, g(x) varies from 1 to 2. For π/4 ≤ x < π, g(x) = 3 - cot x. As cot x varies from 1 to -∞ in this interval, g(x) varies from 2 to ∞. Therefore, the range of g(x) includes values greater than 2. The composition f(g(x)) is defined as follows:

$$f(g(x)) = \begin{cases} (1 + \tan x) + 2, & 0 \le x < \frac{\pi}{4} \\ 6 - (3 - \cot x), & \frac{\pi}{4} \le x < \pi \end{cases}$$

This simplifies to:

$$f(g(x)) = \begin{cases} 3 + \tan x, & 0 \le x < \frac{\pi}{4} \\ 3 + \cot x, & \frac{\pi}{4} \le x < \pi \end{cases}$$

f(g(x)) is continuous in the interval [0, π/4) because it is a sum of continuous functions. f(g(x)) is also continuous in the interval (π/4, π) for the same reason. At x = π/4, the left-hand limit is lim (x→(π/4)-) f(g(x)) = 3 + tan(π/4) = 3 + 1 = 4. The right-hand limit is lim (x→(π/4)+) f(g(x)) = 3 + cot(π/4) = 3 + 1 = 4. The function value at x = π/4 is f(g(π/4)) = 3 + cot(π/4) = 4. Since the left-hand limit, the right-hand limit, and the function value are all equal, f(g(x)) is continuous at x = π/4. Therefore, f(g(x)) is continuous in the interval [0, π)**

Examining the Differentiability of f(g(x))

Now, let's turn our attention to the differentiability of f(g(x)). For a composite function to be differentiable at a point, both the individual functions and the composite function itself must be continuous and have derivatives at that point. We have already established the continuity of f(g(x)) in the interval [0, π).

To determine the differentiability, we need to find the derivative of f(g(x)). Using the chain rule, we first express f(g(x)) as piecewise functions:

$$f(g(x)) = \begin{cases} 3 + \tan x, & 0 \le x < \frac{\pi}{4} \\ 3 + \cot x, & \frac{\pi}{4} \le x < \pi \end{cases}$$

Now, we find the derivative of each piece:

For 0 ≤ x < π/4, the derivative is:

$$\frac{d}{dx}(3 + \tan x) = \sec^2 x$$

For π/4 < x < π, the derivative is:

$$\frac{d}{dx}(3 + \cot x) = -\csc^2 x$$

Thus, the derivative of f(g(x)) is:

$$f'(g(x)) = \begin{cases} \sec^2 x, & 0 \le x < \frac{\pi}{4} \\ -\csc^2 x, & \frac{\pi}{4} < x < \pi \end{cases}$$

We need to check the differentiability at x = π/4. To do this, we compare the left-hand derivative and the right-hand derivative at this point.

The left-hand derivative at x = π/4 is:

$$\lim_{x \to (\pi/4)^-} \sec^2 x = \sec^2(\frac{\pi}{4}) = (\sqrt{2})^2 = 2$$

The right-hand derivative at x = π/4 is:

$$\lim_{x \to (\pi/4)^+} -\csc^2 x = -\csc^2(\frac{\pi}{4}) = -(\sqrt{2})^2 = -2$$

Since the left-hand derivative (2) is not equal to the right-hand derivative (-2), f(g(x)) is not differentiable at x = π/4. Therefore, **f(g(x)) is not differentiable in the interval [0, π)****.

Conclusion

In summary, we have analyzed the continuity and differentiability of the composite function f(g(x)) within the interval [0, π). We found that f(g(x)) is continuous in this interval because both f(x) and g(x) are continuous in their respective domains, and the composition maintains continuity. However, f(g(x)) is not differentiable at x = π/4 due to the mismatch between the left-hand and right-hand derivatives at this point. Therefore, f(g(x)) is not differentiable in the interval [0, π). This analysis highlights the importance of considering both continuity and differentiability when dealing with composite functions, and it demonstrates how the properties of individual functions can affect the properties of their composition. Understanding these concepts is essential for advanced mathematical analysis and applications in various scientific and engineering fields.

• Continuity
• Differentiability
• Composite function
• Piecewise function
• Left-hand limit
• Right-hand limit
• Derivatives
• Chain rule
• Trigonometric functions