If `f(x) = (1)/(x^(2) - 17 x + 66)`, then `f((2)/(x - 2))` is discontinuous at x =

Options

A. 2
B. `(7)/(3)`
C. `(24)/(11)`
D. 6, 11

Answer

To determine where the function \( f\left(\frac{2}{x-2}\right) \) is discontinuous, we will follow these steps:

### Step 1: Identify the points of discontinuity for \( f(x) \)

The function \( f(x) = \frac{1}{x^2 - 17x + 66} \) is discontinuous where the denominator is equal to zero. We first need to find the roots of the quadratic equation:

\[ x^2 - 17x + 66 = 0 \]

### Step 2: Factor the quadratic equation

To factor the quadratic, we look for two numbers that multiply to \( 66 \) and add up to \( -17 \). The factors of \( 66 \) that satisfy this condition are \( -11 \) and \( -6 \). Thus, we can factor the equation as follows:

\[ (x - 11)(x - 6) = 0 \]

### Step 3: Solve for \( x \)

Setting each factor to zero gives us the points of discontinuity for \( f(x) \):

\[ x - 11 = 0 \quad \Rightarrow \quad x = 11 \]
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]

So, \( f(x) \) is discontinuous at \( x = 11 \) and \( x = 6 \).

### Step 4: Determine where \( f\left(\frac{2}{x-2}\right) \) is discontinuous

Next, we need to find where \( \frac{2}{x-2} \) equals \( 11 \) and \( 6 \) since those are the points where \( f(x) \) is discontinuous.

1. Set \( \frac{2}{x-2} = 11 \):

\[ 2 = 11(x - 2) \quad \Rightarrow \quad 2 = 11x - 22 \quad \Rightarrow \quad 11x = 24 \quad \Rightarrow \quad x = \frac{24}{11} \]

2. Set \( \frac{2}{x-2} = 6 \):

\[ 2 = 6(x - 2) \quad \Rightarrow \quad 2 = 6x - 12 \quad \Rightarrow \quad 6x = 14 \quad \Rightarrow \quad x = \frac{7}{3} \]

### Step 5: Identify additional discontinuity from the transformation

The function \( f\left(\frac{2}{x-2}\right) \) will also be discontinuous at \( x = 2 \) because the expression \( \frac{2}{x-2} \) is undefined when \( x = 2 \).

### Conclusion

Thus, the points of discontinuity for \( f\left(\frac{2}{x-2}\right) \) are:

1. \( x = 2 \)
2. \( x = \frac{24}{11} \)
3. \( x = \frac{7}{3} \)

### Final Answer

The function \( f\left(\frac{2}{x-2}\right) \) is discontinuous at \( x = 2, \frac{24}{11}, \frac{7}{3} \).

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