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!