y=(3x-2)/[(x-5)(x-7)] in interval notation

To express the domain of the function \( y = \frac{3x - 2}{(x - 5)(x - 7)} \) in interval notation, we need to identify any values of \( x \) that make the denominator zero. These are the values where the function is undefined.

1. Set the denominator equal to zero:
\[
(x - 5)(x - 7) = 0
\]
This gives us two solutions:
\[
x = 5 \quad \text{and} \quad x = 7
\]

Thus, the function is undefined at \( x = 5 \) and \( x = 7 \).

2. The domain of the function includes all real numbers except \( x = 5 \) and \( x = 7 \). In interval notation, this is written as:
\[
(-\infty, 5) \cup (5, 7) \cup (7, \infty)
\]

This represents the complete domain where the function is defined.

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Would you like more details on this or have any questions?

Here are 5 related questions for further exploration:
1. How do you find vertical asymptotes for rational functions like this one?
2. What happens to the function as \( x \) approaches 5 or 7?
3. How do you determine the horizontal asymptote of this function?
4. What is the behavior of the function for very large or very small values of \( x \)?
5. How do you find the critical points of this function?

**Tip:** Always check the denominator of a rational function to find values that should be excluded from its domain!

y = (3x - 2) / [(x - 5)(x - 7)] in interval notation

Explore how to determine the domain of the rational function y = (3x - 2) / [(x - 5)(x - 7)] using interval notation. This solution identifies the values of x that make the denominator zero and expresses the domain in proper format. Ideal for students in Grades 10-12.

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