is f(x) = 6/x^2-3x-10 discontinuous at x = 0


The function \( f(x) = \frac{6}{x^2 - 3x - 10} \) can potentially have discontinuities where the denominator is equal to zero, as division by zero is undefined.

To check where the function is discontinuous, we need to find where the denominator \( x^2 - 3x - 10 = 0 \).

### Step 1: Factor the denominator
We solve for \( x \) in the quadratic equation \( x^2 - 3x - 10 = 0 \).

Factor the quadratic:
\[
x^2 - 3x - 10 = (x - 5)(x + 2) = 0
\]

### Step 2: Find where the denominator is zero
Now set each factor equal to zero:
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]

### Conclusion
The function is discontinuous at \( x = 5 \) and \( x = -2 \), as these values make the denominator zero. However, \( x = 0 \) is not one of these values, so the function is continuous at \( x = 0 \).

Would you like further clarification or details?

Here are five related questions to consider:
1. How do you identify discontinuities in a rational function?
2. What are the types of discontinuities a function can have?
3. How can you factor more complex quadratic expressions?
4. Can functions have points of continuity but be undefined elsewhere?
5. What is the difference between removable and non-removable discontinuities?

**Tip:** Always check the domain of a rational function by factoring the denominator, as discontinuities occur where it equals zero.

is f(x) = 6/x^2-3x-10 discontinuous at x = 0

Explore how to find discontinuities in the rational function f(x) = 6/(x^2 - 3x - 10). Learn key algebraic concepts like factoring quadratic equations to determine where the function is undefined and continuous. Suitable for Grades 10-12 students.

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