find the x and y intercept of funtion f(x)=2x+3/x-1

To find the intercepts of the function \( f(x) = \frac{2x + 3}{x - 1} \), we'll calculate both the **x-intercept** and the **y-intercept**:

### 1. **Finding the x-intercept:**

The x-intercept occurs when \( f(x) = 0 \). This happens when the numerator of the function is equal to zero (since the denominator would make the function undefined if it were zero).

\[
\frac{2x + 3}{x - 1} = 0
\]

This implies that:

\[
2x + 3 = 0
\]

Solving for \( x \):

\[
2x = -3
\]

\[
x = -\frac{3}{2}
\]

Thus, the **x-intercept** is at \( \left( -\frac{3}{2}, 0 \right) \).

### 2. **Finding the y-intercept:**

The y-intercept occurs when \( x = 0 \). We substitute \( x = 0 \) into the function to find \( f(0) \):

\[
f(0) = \frac{2(0) + 3}{0 - 1} = \frac{3}{-1} = -3
\]

Thus, the **y-intercept** is at \( (0, -3) \).

### Summary of intercepts:
- **x-intercept**: \( \left( -\frac{3}{2}, 0 \right) \)
- **y-intercept**: \( (0, -3) \)

Would you like more details or further clarification on this?

#### Related questions:
1. How do vertical asymptotes relate to the denominator in rational functions?
2. Can we find the horizontal asymptote of this function?
3. How would the graph of \( f(x) = \frac{2x+3}{x-1} \) behave near \( x = 1 \)?
4. What happens to the function as \( x \to \infty \)?
5. How do we find the slope of the tangent line at specific points on this function?

#### Tip:
When solving for intercepts in rational functions, always check if the denominator could be zero to avoid undefined values.

find the x and y intercept of function f(x)=2x+3/x-1

Learn how to find the x and y intercepts of the rational function f(x) = (2x + 3)/(x - 1). This problem involves algebraic manipulation of rational functions, with a detailed step-by-step guide suitable for students in Grades 10-12.

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