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The function given is \( f(x) = \frac{9}{1 + e^{5x}} \).

To locate the discontinuities of this function, we need to check for points where the function is either not defined or not continuous. Discontinuities in rational functions (fractions) generally occur where the denominator is zero. Therefore, we look at the denominator of the function:

\[ 1 + e^{5x} = 0 \]

However, \( e^{5x} \) is an exponential function, and since the exponential function is always positive for any real \( x \), \( 1 + e^{5x} \) can never be zero. This implies that \( 1 + e^{5x} \) is always positive and never zero.

Therefore, the function \( f(x) = \frac{9}{1 + e^{5x}} \) has no points where the denominator equals zero and hence no points of discontinuity. The function is continuous for all real numbers \( x \).

If you need further clarification or details, feel free to ask!

Would you like to explore any of these related topics?
1. What are removable and non-removable discontinuities?
2. How to analyze the behavior of \( f(x) \) as \( x \) approaches infinity?
3. Why is the exponential function always positive?
4. What would make a function like \( f(x) \) have asymptotes?
5. How can exponential functions be transformed and what effects do these transformations have on their graphs?

**Tip:** Remember, in mathematics, understanding the behavior of functions at extreme values of \( x \) (like infinity or negative infinity) can provide deep insights into their overall characteristics.

Locate the discontinuities of the function f(x) = 9 / (1 + e^(5/x)).

Analyze the discontinuities of the function f(x) = 9 / (1 + e^(5/x)). This problem explores exponential functions and rational discontinuities, suitable for students in Grades 11-12 or university level calculus.

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