Questions: f(x)=(2x-3)/(x+4)

f(x)=(2x-3)/(x+4)
Transcript text: $f(x)=\frac{2 x-3}{x+4}$
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Solution

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Solution Steps

To analyze the function \( f(x) = \frac{2x - 3}{x + 4} \), we can:

  1. Identify the domain by finding values of \( x \) that make the denominator zero.
  2. Determine the vertical asymptote by setting the denominator equal to zero.
  3. Find the horizontal asymptote by analyzing the behavior of the function as \( x \) approaches infinity.
Step 1: Determine the Domain

The function \( f(x) = \frac{2x - 3}{x + 4} \) is defined for all \( x \) except where the denominator is zero. Setting the denominator equal to zero gives:

\[ x + 4 = 0 \implies x = -4 \]

Thus, the domain of \( f(x) \) is all real numbers except \( -4 \):

\[ \text{Domain: } x \in \mathbb{R}, x \neq -4 \]

Step 2: Identify the Vertical Asymptote

The vertical asymptote occurs where the denominator is zero. From the previous calculation, we find:

\[ \text{Vertical Asymptote: } x = -4 \]

Step 3: Find the Horizontal Asymptote

To find the horizontal asymptote, we analyze the behavior of \( f(x) \) as \( x \) approaches infinity. The leading coefficients of the numerator and denominator are both 2 and 1, respectively. Therefore, the horizontal asymptote is given by:

\[ \text{Horizontal Asymptote: } y = \frac{2}{1} = 2 \]

Final Answer

The results are summarized as follows:

  • Domain: \( x \in \mathbb{R}, x \neq -4 \)
  • Vertical Asymptote: \( x = -4 \)
  • Horizontal Asymptote: \( y = 2 \)

Thus, the final boxed answers are:

\[ \boxed{\text{Domain: } x \in \mathbb{R}, x \neq -4} \] \[ \boxed{\text{Vertical Asymptote: } x = -4} \] \[ \boxed{\text{Horizontal Asymptote: } y = 2} \]

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