The function `f(x)=(16+x^(2))/(9-x^(2))` is :

Options

A. discontinuous at only one point.
B. discontinuous at exactly two points.
C. discontinuous at exactly three points.
D. a continuos function.

Answer

To determine the nature of the function \( f(x) = \frac{16 + x^2}{9 - x^2} \), we need to analyze its discontinuities. Here’s a step-by-step solution:

### Step 1: Identify the function
The function is given as:
\[ f(x) = \frac{16 + x^2}{9 - x^2} \]

### Step 2: Determine when the function is discontinuous
A rational function \( f(x) = \frac{g(x)}{h(x)} \) is discontinuous when the denominator \( h(x) \) is equal to zero. In our case, the denominator is:
\[ h(x) = 9 - x^2 \]

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

### Step 4: Solve for \( x \)
Rearranging the equation gives:
\[ x^2 = 9 \]
Taking the square root of both sides, we find:
\[ x = 3 \quad \text{or} \quad x = -3 \]

### Step 5: Conclusion about discontinuity
The function \( f(x) \) is discontinuous at the points \( x = 3 \) and \( x = -3 \). Therefore, there are two points of discontinuity.

### Step 6: Identify the correct option
Based on the analysis, we conclude that the correct option is that the function is discontinuous at two points: \( x = -3 \) and \( x = 3 \).

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