Where does the reciprocal function of f(x)=3 - x increase?
a.	{x ∈ R | x > 3 }	   c.	{x ∈ R | x ≠ 3}
b.	{x ∈ R | x < 3 }	   d.	{x ∈ R }

To determine where the reciprocal function of \( f(x) = 3 - x \) increases, we first need to define the reciprocal function. 

The reciprocal function of \( f(x) = 3 - x \) is:

\[ g(x) = \frac{1}{f(x)} = \frac{1}{3 - x} \]

Next, we need to find the interval where \( g(x) \) is increasing. To do this, we consider the derivative of \( g(x) \) with respect to \( x \).

\[ g(x) = \frac{1}{3 - x} \]

We use the quotient rule to find the derivative of \( g(x) \):

\[ g'(x) = \frac{0(3 - x) - 1(-1)}{(3 - x)^2} = \frac{1}{(3 - x)^2} \]

The derivative \( g'(x) = \frac{1}{(3 - x)^2} \) is always positive for all \( x \) except at \( x = 3 \) where it is undefined (since the denominator becomes zero and division by zero is undefined).

Since \( g'(x) \) is positive, \( g(x) = \frac{1}{3 - x} \) is an increasing function for all \( x \) where \( g(x) \) is defined.

Therefore, the correct answer is:

c. \(\{x \in \mathbb{R} \mid x \neq 3\}\)

Would you like more details or have any questions? Here are 5 more questions you might be interested in:

1. How do you find the domain of a function?
2. What is the derivative of a rational function?
3. How can you determine if a function is increasing or decreasing on an interval?
4. What are the properties of reciprocal functions?
5. How do you find the critical points of a function?

**Tip:** To determine where a function is increasing or decreasing, finding and analyzing its derivative is a very effective method.

Explore where the reciprocal function of f(x) = 3 - x increases with a detailed step-by-step solution. Learn about reciprocal functions, derivatives, and how to determine where a function is increasing or decreasing. Suitable for high school students.

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