Question no 49the function f : r. r, is defined as f(x) = x²-6x+10/3x-3-x² is

To analyze the function \( f(x) = \frac{x^2 - 6x + 10}{3x - 3 - x^2} \), we need to break it down step by step. This function is a rational function, meaning it is the ratio of two polynomials. Let's explore its components and behavior.

Understanding the Components

The function consists of a numerator and a denominator:

• Numerator: \( x^2 - 6x + 10 \)
• Denominator: \( 3x - 3 - x^2 \)

Analyzing the Numerator

The numerator is a quadratic expression. We can rewrite it in vertex form to understand its properties better. Completing the square:

\( x^2 - 6x + 10 = (x - 3)^2 + 1 \)

This shows that the parabola opens upwards, has a vertex at \( (3, 1) \), and does not cross the x-axis, indicating it is always positive.

Examining the Denominator

Now, let's look at the denominator:

\( 3x - 3 - x^2 = -x^2 + 3x - 3 \)

Rearranging gives us \( -(x^2 - 3x + 3) \). To find its roots, we can use the quadratic formula:

\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( a = -1 \), \( b = 3 \), and \( c = -3 \). Plugging in these values:

\( x = \frac{-3 \pm \sqrt{3^2 - 4(-1)(-3)}}{2(-1)} = \frac{-3 \pm \sqrt{9 - 12}}{-2} = \frac{-3 \pm \sqrt{-3}}{-2} \)

Since the discriminant is negative, the denominator has no real roots, meaning it does not cross the x-axis and is always negative.

Behavior of the Function

Now that we understand both parts, we can analyze the overall function:

• The numerator is always positive.
• The denominator is always negative.

Thus, the function \( f(x) \) will always be negative for all real values of \( x \). This means that the graph of the function will lie below the x-axis.

Finding Asymptotes

Since the denominator does not equal zero for any real \( x \), there are no vertical asymptotes. However, we can consider horizontal asymptotes by examining the degrees of the polynomials:

Both the numerator and denominator are quadratic, so we can find the horizontal asymptote by taking the leading coefficients:

\( y = \frac{1}{-1} = -1 \)

This indicates that as \( x \) approaches infinity or negative infinity, \( f(x) \) approaches -1.

Summary of Key Points

• The function \( f(x) = \frac{x^2 - 6x + 10}{3x - 3 - x^2} \) is always negative.
• It has no real roots and no vertical asymptotes.
• There is a horizontal asymptote at \( y = -1 \).

In conclusion, this function exhibits interesting behavior as a rational function, and understanding its components helps us predict its overall characteristics. If you have any further questions or need clarification on any part, feel free to ask!