Question

use the graph shown to find the following.
(a) the domain and range of the function
(b) the intercepts, if any
(c) horizontal asymptotes, if any
(d) vertical asymptotes, if any
(e) oblique asymptotes, if any

(a) what is the domain? select the correct choice below and fill in any answer boxes within your choice.
○ a. the domain of the function is $\\{x \mid x \
eq \square\\}$.
(type an inequality in the form $x \
eq$. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)
○ b. the domain of the function in the graph is the set of all real numbers.

what is the range? select the correct choice below and fill in any answer boxes within your choice.
○ a. the range of the function is $\\{y \mid y \
eq \square\\}$
(type an inequality in the form $y \
eq$. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)
○ b. the range of the function in the graph is the set of all real numbers.

(b) find the x-intercepts, if there are any. select the correct choice below and fill in any answer boxes within your choice

Explanation:

Step1: Identify domain from graph

The graph has a vertical asymptote at $x=5$, so $x$ cannot equal 5.
Domain: $\{x \mid x
eq 5\}$

Step2: Identify range from graph

The graph has a horizontal asymptote at $y=5$, so $y$ cannot equal 5.
Range: $\{y \mid y
eq 5\}$

Step3: Find intercepts

The graph crosses the x-axis at $(2,0)$, so this is the x-intercept. There is no y-intercept (since $x=0$ would be a valid input but the graph does not cross the y-axis, and the asymptote doesn't affect this; the only x-intercept is $(2,0)$).

Step4: Identify horizontal asymptote

The graph approaches the horizontal line $y=5$ but never touches it.
Horizontal asymptote: $y=5$

Step5: Identify vertical asymptote

The graph approaches the vertical line $x=5$ but never touches it.
Vertical asymptote: $x=5$

Step6: Check for oblique asymptotes

There is no slanted line that the graph approaches, so no oblique asymptotes.

Answer:

(a) A. The domain of the function is $\{x \mid x
eq 5\}$
(b) x-intercept: $(2,0)$; no y-intercept
(c) Horizontal asymptote: $y=5$
(d) Vertical asymptote: $x=5$
(e) No oblique asymptotes