The function `f (x) = (2-x)/(9x-x^(3))` is:

Options

A. discontinuous at one point
B. discontinuous at two points
C. discontinuous at three points
D. discontinuous at four points

Answer

To determine the points of discontinuity for the function \( f(x) = \frac{2 - x}{9x - x^3} \), we need to analyze the denominator, as a function is discontinuous where its denominator is equal to zero.

### Step-by-Step Solution:

1. **Identify the Denominator**:
The denominator of the function is \( h(x) = 9x - x^3 \).

2. **Set the Denominator to Zero**:
To find the points of discontinuity, we set the denominator equal to zero:
\[ 9x - x^3 = 0 \]

3. **Factor the Denominator**:
We can factor out \( x \) from the equation:
\[ x(9 - x^2) = 0 \]
This gives us one factor \( x = 0 \).

4. **Solve for the Remaining Factors**:
Now, we need to solve \( 9 - x^2 = 0 \):
\[ 9 = x^2 \implies x^2 = 9 \implies x = 3 \quad \text{or} \quad x = -3 \]

5. **List All Points of Discontinuity**:
The points where the function is discontinuous are:
\[ x = 0, \quad x = 3, \quad x = -3 \]

6. **Count the Points of Discontinuity**:
We have found three points of discontinuity: \( x = 0, -3, 3 \).

### Conclusion:
The function \( f(x) \) is discontinuous at three points: \( x = 0, -3, \) and \( 3 \).

### Final Answer:
The function \( f(x) = \frac{2 - x}{9x - x^3} \) is discontinuous at **3 points**.

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