7. the graph of f(x) is shown below.
the domain of this function is
(1) 0,12
(2) 15,45
(3) 0

Question

7. the graph of f(x) is shown below.
the domain of this function is
(1) 0,12
(2) 15,45
(3) 0 < x ≤ 12
(4) 15 < x ≤ 45
8. a child tosses a baseball up into the air. on its way down, it gets caught in a tree for several seconds before falling back down to the ground.
select the best description of the range of this function.
a. the range includes all numbers from 5 to 12.5.
b. the range includes all integers between 0 and 12.5.
c. the range includes all numbers from 0 to 8.5.
d. the range includes all numbers from 0 to 12.5.
7. the graph of h shows the height, in meters, of a rocket t seconds after it was launched.
a. find h(0). what does this value represent?
b. describe the domain of this function.
c. describe the range of this function.
solve h(x)=0. what does this value represent?

Accepted Answer

### Problem 7 (Top - Graph of $ f(x) $)
# Explanation:
## Step 1: Recall Domain Definition
The domain of a function is the set of all possible $ x $-values (input values) for which the function is defined. We analyze the graph's $ x $-axis.
## Step 2: Analyze the Graph's $ x $-Intervals
- The graph starts at an open circle at $ x = 0 $ (so $ x>0 $) and ends at a closed dot (so $ x\leq12 $) (by observing the $ x $-axis, the last point is around $ x = 12 $). So the domain is $ 0 < x\leq12 $, which is option (3).
# Answer:
(3) $ 0 < x\leq12 $

### Problem 8 (Baseball Toss)
# Brief Explanations:
The range of a function is the set of all possible $ y $-values (output values). The graph's $ y $-axis (height) starts at 0 (when time is 0, height is 0) and goes up to a maximum around 12.5. Option A is wrong (starts at 5), B is wrong (range includes non - integers), C is wrong (max is higher than 8.5). So D is correct.
# Answer:
D. The range includes all numbers from 0 to 12.5.

### Problem 7 (Rocket Height)
#### Part a: Find $ H(0) $
# Explanation:
## Step 1: Interpret $ H(0) $
$ H(0) $ is the value of the function $ H $ when $ t = 0 $ (time is 0 seconds, at launch). From the graph, when $ t = 0 $, the height ( $ y $-value) is 15 meters (by looking at the $ y $-axis intercept).
## Step 2: State the Representation
$ H(0)=15 $ meters. This value represents the height of the rocket at the moment it was launched ( $ t = 0 $ seconds).
# Answer:
$ H(0)=15 $; it represents the rocket's height at launch ( $ t = 0 $ seconds).

#### Part b: Domain of the Function
# Explanation:
## Step 1: Recall Domain Definition
The domain is the set of all possible $ t $-values (time in seconds).
## Step 2: Analyze the Graph's $ t $-Intervals
The graph starts at $ t = 0 $ (launch) and ends at $ t = 5 $ seconds (when the rocket lands, height is 0). So the domain is all real numbers $ t $ such that $ 0\leq t\leq5 $.
# Answer:
The domain is $ 0\leq t\leq5 $ (all real numbers from 0 to 5 seconds, inclusive).

#### Part c: Range of the Function
# Explanation:
## Step 1: Recall Range Definition
The range is the set of all possible height values ( $ y $-values).
## Step 2: Analyze the Graph's $ y $-Intervals
The minimum height is 0 meters (when the rocket lands at $ t = 5 $) and the maximum height is the peak of the parabola - shaped graph. From the graph, the minimum height is 0 meters and the maximum height is 35 meters (approximate from the $ y $-axis). So the range is all real numbers from 0 to 35 meters (inclusive), or more precisely, looking at the graph, the height starts at 15 meters (wait, no, at $ t = 0 $, height is 15, then goes up to a peak and down to 0. Wait, re - analyzing: when $ t = 0 $, height is 15, then it goes up to a maximum (let's say around 35) and then down to 0 at $ t = 5 $. So the range is all real numbers $ h $ such that $ 0\leq h\leq35 $ (or more accurately, from the graph's $ y $-values: the lowest $ y $ is 0, the highest is the peak, and the starting point is 15, but since it goes from 0 (at $ t = 5 $) up to 15 (at $ t = 0 $) then up to a peak and down to 0? Wait, no, the graph starts at $ (0,15) $, goes up to a peak, then down to $ (5,0) $. So the minimum height is 0 (at $ t = 5 $) and the maximum height is the peak (let's say 35). So the range is $ 0\leq h\leq35 $ (all real numbers from 0 to 35 meters, inclusive).
# Answer:
The range is all real numbers from 0 to 35 meters (inclusive), representing all possible heights the rocket reaches during its flight.

#### Part d: Solve $ H(x)=0 $
# Explanation:
## Step 1: Interpret $ H(x) = 0 $
$ H(x)=0 $ means we find the time $ t $ ( $ x $ - value) when the height of the rocket is 0 meters (it lands). From the graph, when $ y = 0 $, $ t = 5 $ seconds.
## Step 2: State the Representation
The solution to $ H(x)=0 $ is $ x = 5 $ seconds. This value represents the time at which the rocket lands (when its height is 0 meters).
# Answer:
$ x = 5 $; it represents the time when the rocket lands (height is 0 meters).

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QUESTION IMAGE

Question

Explanation:

Problem 7 (Top - Graph of \( f(x) \))

Step 1: Recall Domain Definition

Step 2: Analyze the Graph's \( x \)-Intervals

Step 1: Interpret \( H(0) \)

Step 2: State the Representation

Answer:

Problem 8 (Baseball Toss)