Question

Explain how the graph of the function ​f(x)equalsStartFraction 1 Over x plus 3 EndFraction
can be obtained from the graph of yequalsStartFraction 1 Over x EndFraction
. Then graph f and give the​ (a) domain and​ (b) range. Determine the largest open intervals of the domain over which the function is​ (c) increasing or​ (d) decreasing.
Question content area bottom
Part 1
To obtain the graph of​ f, shift the graph of yequalsStartFraction 1 Over x EndFraction

▼
down
to the right
to the left
up

  
enter your response here ​unit(s).
Part 2
Graph the function ​f(x)equalsStartFraction 1 Over x plus 3 EndFraction
. Choose the correct graph below.
A.
-16
16
-16
16
x
y



A coordinate system has a horizontal x-axis labeled from negative 16 to 16 in increments of 2 and a vertical y-axis labeled from negative 16 to 16 in increments of 2. A graph has two branches. The first branch is below y=3 and to the left of the y-axis, approaching both. The second branch is above y=3 and to the right of the y-axis, approaching both.
B.
-16
16
-16
16
x
y



A coordinate system has a horizontal x-axis labeled from negative 16 to 16 in increments of 2 and a vertical y-axis labeled from negative 16 to 16 in increments of 2. A graph has two branches. The first branch is below y= negative 3 and to the left of the y-axis, approaching both. The second branch is above y=negative 3 and to the right of the y-axis, approaching both.
C.
-16
16
-16
16
x
y



A coordinate system has a horizontal x-axis labeled from negative 16 to 16 in increments of 2 and a vertical y-axis labeled from negative 16 to 16 in increments of 2. A graph has two branches. The first branch is below the x-axis and to the left of x = negative 3, approaching both. The second branch is above the x-axis and to the right of x = negative 3, approaching both.
D.
-16
16
-16
16
x
y



A coordinate system has a horizontal x-axis labeled from negative 16 to 16 in increments of 2 and a vertical y-axis labeled from negative 16 to 16 in increments of 2. A graph has two branches. The first branch is below the x-axis and to the left of x=3, approaching both. The second branch is above the x-axis and to the right of x=3, approaching both.
Part 3
​(a) The domain of​ f(x) is
  
enter your response here.
​(Type your answer in interval​ notation.)
Part 4
​(b) The range of​ f(x) is
  
enter your response here.
​(Type your answer in interval​ notation.)
Part 5
​(c) Determine the largest open intervals of the domain over which the function is increasing. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The function is increasing on
  
enter your response here.
​(Type your answer in interval notation. Use a comma to separate answers as​ needed.)
B.
There is no interval of the domain for which the function is increasing.
Part 6
​(d) Determine the largest open intervals of the domain over which the function is decreasing. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The function is decreasing on
  
enter your response here.
​(Type your answer in interval notation. Use a comma to separate answers as​ needed.)
B.
There is no interval of the domain for which the function is decreasing.
To obtain the graph of f, shift the graph of yStartFraction 1 Over x EndFractionDetermine the largest open intervals of the domain over which the function is (c) increasing or (d) decreasing.unit(s).Choose the correct graph below.(a) The domain of f(x) is(Type your answer in interval notation.)(b) The range of f(x) is(Type your answer in interval notation.))Select the correct choice below and, if necessary, fill in the answer box to complete your choice.input field 5Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Answer 1

Part 1 — transformation
- f(x) = 1/(x+3) is obtained from y = 1/x by shifting the graph left 3 units.

Part 2 — graph
- The correct graph is C (vertical asymptote at x = −3, horizontal asymptote y = 0; branch below the x-axis for x < −3 and above the x-axis for x > −3).

Part 3 — domain
- Domain: (-∞, -3) ∪ (-3, ∞)

Part 4 — range
- Range: (-∞, 0) ∪ (0, ∞)

Part 5 — increasing
- There is no interval where f is increasing. (f′(x) = −1/(x+3)^2 < 0 on its domain.)

Part 6 — decreasing
- The function is decreasing on (-∞, -3) and on (-3, ∞).