Question

sketch the asymptotes and the graph of the function. identify the domain and range.
$y = \frac{1}{x - 4} + 3$

Explanation:

Step1: Find Vertical Asymptote

For a rational function \( y = \frac{1}{x - 4}+3 \), the vertical asymptote occurs where the denominator is zero. Set \( x - 4 = 0 \), so \( x = 4 \).

Step2: Find Horizontal Asymptote

For a function of the form \( y=\frac{a}{x - h}+k \), the horizontal asymptote is \( y = k \). Here, \( k = 3 \), so horizontal asymptote is \( y = 3 \).

Step3: Determine Domain

The domain is all real numbers except where the function is undefined (where denominator is zero). So domain is \( \{x|x
eq4, x\in\mathbb{R}\} \).

Step4: Determine Range

Since \( \frac{1}{x - 4}\) can never be zero, \( y=\frac{1}{x - 4}+3 \) can never be 3. So range is \( \{y|y
eq3, y\in\mathbb{R}\} \).

Step5: Sketch the Graph

• The graph of \( y=\frac{1}{x} \) is a hyperbola.
• For \( y=\frac{1}{x - 4}+3 \), we shift the graph of \( y = \frac{1}{x} \) 4 units to the right (because of \( x - 4 \)) and 3 units up (because of \( + 3 \)). The two branches of the hyperbola approach the asymptotes \( x = 4 \) and \( y = 3 \).

Answer:

• Vertical Asymptote: \( x = 4 \)
• Horizontal Asymptote: \( y = 3 \)
• Domain: All real numbers except \( x = 4 \) (or \( (-\infty, 4)\cup(4, \infty) \))
• Range: All real numbers except \( y = 3 \) (or \( (-\infty, 3)\cup(3, \infty) \))
• Graph: A hyperbola shifted 4 units right and 3 units up from \( y=\frac{1}{x} \), approaching asymptotes \( x = 4 \) and \( y = 3 \).