Questions: f(x)=(3x-8)/(x^2-9x+20)

Transcript text: $f(x)=\frac{3 x-8}{x^{2}-9 x+20}$

Solution

Solution Steps

To analyze the function $ f(x) = \frac{3x - 8}{x^2 - 9x + 20} $, we need to:

Identify the domain of the function by finding the values of $ x $ that make the denominator zero.
Simplify the function if possible.
Determine the vertical and horizontal asymptotes.

Step 1: Identify the Domain

To find the domain of $ f(x) = \frac{3x - 8}{x^2 - 9x + 20} $, we need to determine the values of $ x $ that make the denominator zero. Solving $ x^2 - 9x + 20 = 0 $ gives: \[ x = 4 \quad \text{and} \quad x = 5 \] Thus, the domain of $ f(x) $ is all real numbers except $ x = 4 $ and $ x = 5 $.

Step 2: Simplify the Function

The function $ f(x) $ cannot be simplified further as the numerator and the denominator do not have common factors.

Step 3: Determine Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is non-zero. From the previous step, we know the vertical asymptotes are at: \[ x = 4 \quad \text{and} \quad x = 5 \]

Step 4: Determine Horizontal Asymptote

To find the horizontal asymptote, we compare the degrees of the numerator and the denominator. Since the degree of the numerator (1) is less than the degree of the denominator (2), the horizontal asymptote is: \[ y = 0 \]

Final Answer

Domain: All real numbers except $ x = 4 $ and $ x = 5 $
Vertical Asymptotes: $ x = 4 $ and $ x = 5 $
Horizontal Asymptote: $ y = 0 $

\[ \boxed{ \begin{aligned} &\text{Domain: } \mathbb{R} \setminus \{4, 5\} \\ &\text{Vertical Asymptotes: } x = 4, x = 5 \\ &\text{Horizontal Asymptote: } y = 0 \end{aligned} } \]

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