Determine the domain and range of the following functions

(A) y = tanx

(B)y = 1/(x(x - 1))



Let's determine the domain and range of each function.

### (A) \( y = \tan(x) \)

**Domain**:  
The tangent function, \( \tan(x) \), is undefined where the cosine function \( \cos(x) \) is zero. This occurs at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is any integer.

Thus, the domain of \( y = \tan(x) \) is:
\[ x \in \mathbb{R} \setminus \left\{ \frac{\pi}{2} + n\pi \mid n \in \mathbb{Z} \right\} \]
Or in interval notation:
\[ x \in \left( -\infty, \frac{\pi}{2} \right) \cup \left( \frac{\pi}{2}, \frac{3\pi}{2} \right) \cup \left( \frac{3\pi}{2}, \frac{5\pi}{2} \right) \cup \dots \]

**Range**:  
The range of \( y = \tan(x) \) is all real numbers since the tangent function can take any real value:
\[ y \in \mathbb{R} \]

### (B) \( y = \frac{1}{x(x - 1)} \)

**Domain**:  
The function is undefined where the denominator is zero, which occurs at \( x = 0 \) and \( x = 1 \).

Thus, the domain of \( y = \frac{1}{x(x - 1)} \) is:
\[ x \in \mathbb{R} \setminus \{0, 1\} \]
Or in interval notation:
\[ x \in (-\infty, 0) \cup (0, 1) \cup (1, \infty) \]

**Range**:  
To determine the range, consider the function's behavior as \( x \) approaches the values that make the denominator zero and the limits as \( x \) approaches infinity. The function can take on all real values except \( y = 0 \) because there's no \( x \) that makes \( \frac{1}{x(x-1)} = 0 \).

Thus, the range of \( y = \frac{1}{x(x - 1)} \) is:
\[ y \in \mathbb{R} \setminus \{0\} \]

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Would you like further details on any of these points, or do you have any questions?

### Related Questions:
1. What are the domain and range of \( y = \cot(x) \)?
2. How does the domain of \( y = \sec(x) \) compare to \( y = \tan(x) \)?
3. What is the domain and range of \( y = \frac{1}{x^2 + 1} \)?
4. How do asymptotes affect the range of rational functions?
5. Can you explain the domain and range of inverse trigonometric functions?

### Tip:
When determining the domain of a function, always consider the points where the function might be undefined, such as where a denominator equals zero or where a function inside a square root is negative.

Determine the domain and range of the following functions (A) y = tanx (B)y = 1/(x(x - 1))

Learn how to determine the domain and range of functions y = tan(x) and y = 1/(x(x - 1)). This problem involves understanding the properties of trigonometric and rational functions. Suitable for high school students in Grades 11-12.

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