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

Madhav Soni , 8 Years ago
Grade 12
anser 1 Answers
Askiitians Tutor Team

Last Activity: 18 Days ago

To analyze the function f(x)=x26x+103x3x2, 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: x26x+10
  • Denominator: 3x3x2

Analyzing the Numerator

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

x26x+10=(x3)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:

3x3x2=x2+3x3

Rearranging gives us (x23x+3). To find its roots, we can use the quadratic formula:

x=b±b24ac2a

Here, a=1, b=3, and c=3. Plugging in these values:

x=3±324(1)(3)2(1)=3±9122=3±32

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=11=1

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

Summary of Key Points

  • The function f(x)=x26x+103x3x2 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!

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