Is the function: `f(x) = (x^(2) - 8x + 18)/(x^(2) + 4x + 30)` one-one?

Answer

To determine whether the function \( f(x) = \frac{x^2 - 8x + 18}{x^2 + 4x + 30} \) is one-one, we will follow these steps:

### Step 1: Set the function equal for two different inputs
Assume \( f(x_1) = f(x_2) \). This means:

\[ \frac{x_1^2 - 8x_1 + 18}{x_1^2 + 4x_1 + 30} = \frac{x_2^2 - 8x_2 + 18}{x_2^2 + 4x_2 + 30} \]

### Step 2: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:

\[ (x_1^2 - 8x_1 + 18)(x_2^2 + 4x_2 + 30) = (x_2^2 - 8x_2 + 18)(x_1^2 + 4x_1 + 30) \]

### Step 3: Expand both sides
Expanding both sides will lead to a polynomial equation.

For the left side:

\[ x_1^2 x_2^2 + 4x_1^2 x_2 - 8x_1 x_2^2 - 32x_1 + 18x_2^2 + 72x_2 \]

And for the right side:

\[ x_2^2 x_1^2 + 4x_2^2 x_1 - 8x_2 x_1^2 - 32x_2 + 18x_1^2 + 72x_1 \]

### Step 4: Rearranging the equation
Rearranging the equation will lead to:

\[ 0 = (4x_1^2 - 4x_2^2) + (-8x_1 + 8x_2)(x_1^2 + 4x_1 + 30) + (18x_1 - 18x_2)(x_2^2 + 4x_2 + 30) \]

### Step 5: Factor the equation
This can be factored or simplified to find the relationship between \( x_1 \) and \( x_2 \).

### Step 6: Analyze the roots
If we find that there are two distinct values \( x_1 \) and \( x_2 \) such that \( f(x_1) = f(x_2) \) and \( x_1 \neq x_2 \), then the function is not one-one.

### Conclusion
If we can find such values, we conclude that the function \( f(x) \) is not one-one.