Answer:
Vertical asymptote: x = 3
Horizontal asymptote: y = 1
Domain: (-∞, 3) ∪ (3, ∞)
Range: (-∞, 1) ∪ (1, ∞)
x-intercept: (6, 0)
y-intercept: (0, 2)
Step-by-step explanation:
Asymptotes
An asymptote is a line that a curve approaches indefinitely but never touches. Asymptotes represent values where the function is undefined.
Asymptotes are typically shown on a graph as dashed lines. Therefore, the vertical and horizontal asymptotes of the graphed rational function are:
[tex]\Large\boxed{\textsf{Vertical asymptote: $x = 3$}}[/tex]
[tex]\Large\boxed{\textsf{Horizontal asymptote: $y = 1$}}[/tex]
[tex]\dotfill[/tex]
Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined. As there is a vertical asymptote at x = 3, the function is not defined at x = 3. Therefore, the domain of the graphed rational function is:
[tex]\Large\boxed{\textsf{Domain: $(-\infty, 3) \cup (3, \infty)$}}[/tex]
[tex]\dotfill[/tex]
Range
The range of a function is the set of all possible output values (y-values) for which the function is defined. As there is a horizontal asymptote at y = 1, the function can never attain the value y = 1. Therefore, the range of the graphed rational function is:
[tex]\Large\boxed{\textsf{Range: $(-\infty, 1) \cup (1, \infty)$}}[/tex]
[tex]\dotfill[/tex]
x-intercept
The x-intercepts of a function are the points at which the graph of the function intersects the x-axis, so when y = 0.
From observation of the graphed rational function, the curve intersects the x-axis at (6, 0). Therefore, the x-intercept is:
[tex]\Large\boxed{\textsf{$x$-intercept: $(6, 0)$}}[/tex]
[tex]\dotfill[/tex]
y-intercept
The y-intercept of a function is the point at which the graph of the function intersects the y-axis, so when x = 0.
From observation of the graphed rational function, the curve intersects the y-axis at (0, 2). Therefore, the y-intercept is:
[tex]\Large\boxed{\textsf{$y$-intercept: $(0, 2)$}}[/tex]
