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 }

Let's analyze where the reciprocal function of \( f(x) = 3 - x \) increases.

The reciprocal function of \( f(x) \) is:

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

To determine where \( g(x) \) is increasing, let's consider the behavior of \( g(x) \).

We previously found the derivative of \( g(x) \):

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

Since \( (3 - x)^2 \) is always positive for all \( x \) except at \( x = 3 \) where it is undefined, \( g'(x) \) is always positive for \( x \neq 3 \). This means \( g(x) \) is increasing for all \( x \) where it is defined, which is \( x \neq 3 \).

Now, let's match this conclusion with the given options:

a. \(\{ x \in \mathbb{R} \mid x > 3 \}\) - This indicates \( g(x) \) is increasing for \( x \) greater than 3.
b. \(\{ x \in \mathbb{R} \mid x < 3 \}\) - This indicates \( g(x) \) is increasing for \( x \) less than 3.
c. \(\{ x \in \mathbb{R} \mid x \neq 3 \}\) - This indicates \( g(x) \) is increasing for all \( x \) except 3.
d. \(\{ x \in \mathbb{R} \}\) - This indicates \( g(x) \) is increasing for all \( x \).

Since \( g(x) \) is increasing for all \( x \) except \( x = 3 \), 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 determine the intervals where a function is increasing or decreasing?
2. What is the significance of the first derivative test?
3. How do you find the asymptotes of a rational function?
4. Can you explain the concept of critical points in calculus?
5. How do you apply the chain rule to find the derivative of composite functions?

**Tip:** When analyzing the behavior of rational functions, always consider the points where the function is undefined, as these points often influence the overall behavior of the function.

Understand where the reciprocal function of f(x) = 3 - x increases with a detailed analysis. Learn about reciprocal functions, derivatives, and how to determine increasing intervals. Suitable for high school students.

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