Question
How does the domain change from p(x)=−3x−7−−−−√+6
𝑝
(
𝑥
)
=
−
3
𝑥
−
7
+
6
to m(x)=−3x−2−−−−√+6
𝑚
(
𝑥
)
=
−
3
𝑥
−
2
+
6
?(1 point)
Responses
The range will change from [7,∞)
[
7
,
∞
)
to [2,∞)
[
2
,
∞
)
.
The range will change from left bracket 7 comma infinity right parenthesis to left bracket 2 comma infinity right parenthesis .
The domain will not change.
The domain will not change.
The domain will change from [2,∞)
[
2
,
∞
)
to [7,∞)
[
7
,
∞
)
.
The domain will change from left bracket 2 comma infinity right parenthesis to left bracket 7 comma infinity right parenthesis .
The domain will change from [7,∞)
[
7
,
∞
)
to [2,∞)
[
2
,
∞
)
.
𝑝
(
𝑥
)
=
−
3
𝑥
−
7
+
6
to m(x)=−3x−2−−−−√+6
𝑚
(
𝑥
)
=
−
3
𝑥
−
2
+
6
?(1 point)
Responses
The range will change from [7,∞)
[
7
,
∞
)
to [2,∞)
[
2
,
∞
)
.
The range will change from left bracket 7 comma infinity right parenthesis to left bracket 2 comma infinity right parenthesis .
The domain will not change.
The domain will not change.
The domain will change from [2,∞)
[
2
,
∞
)
to [7,∞)
[
7
,
∞
)
.
The domain will change from left bracket 2 comma infinity right parenthesis to left bracket 7 comma infinity right parenthesis .
The domain will change from [7,∞)
[
7
,
∞
)
to [2,∞)
[
2
,
∞
)
.
Answers
Answered by
GPT-4o mini
To analyze the changes in the domain between the functions \( p(x) = -\sqrt{-3x - 7} + 6 \) and \( m(x) = -\sqrt{-3x - 2} + 6 \), we need to determine when the expressions inside the square roots are non-negative, as square roots of negative numbers are not defined in the real number system.
1. **Finding the domain of \( p(x) \):**
\[
-3x - 7 \geq 0
\]
Solving for \( x \):
\[
-3x \geq 7
\]
\[
x \leq -\frac{7}{3}
\]
This means the domain for \( p(x) \) is:
\[
(-\infty, -\frac{7}{3}]
\]
2. **Finding the domain of \( m(x) \):**
\[
-3x - 2 \geq 0
\]
Solving for \( x \):
\[
-3x \geq 2
\]
\[
x \leq -\frac{2}{3}
\]
This means the domain for \( m(x) \) is:
\[
(-\infty, -\frac{2}{3}]
\]
Now we compare the domains:
- The domain of \( p(x) \) is \( (-\infty, -\frac{7}{3}] \).
- The domain of \( m(x) \) is \( (-\infty, -\frac{2}{3}] \).
The key change is that both functions are defined for \( x \) values less than or equal to different cutoff points: \( -\frac{7}{3} \) for \( p(x) \) and \( -\frac{2}{3} \) for \( m(x) \). Therefore, \( m(x) \) includes all the values \( p(x) \) does but more since \( -\frac{2}{3} \) is greater than \( -\frac{7}{3} \).
To summarize, the domain changes as follows:
- The domain of \( p(x) \) is \( (-\infty, -\frac{7}{3}] \).
- The domain of \( m(x) \) is \( (-\infty, -\frac{2}{3}] \).
**Conclusion:** The domain actually goes from \( (-\infty, -\frac{7}{3}] \) for \( p(x) \) to \( (-\infty, -\frac{2}{3}] \) for \( m(x) \), indicating that the domain has become larger.
### Correct Response
The domain will change from \( (-\infty, -\frac{7}{3}] \) to \( (-\infty, -\frac{2}{3}] \).
However, this does not match exactly with your provided response format. If I relate with intervals you are using, for absolute values, that is not precise from above but for square roots function similar properties hold.
If utilizing the response format given in your question aligning with realistic ranges would be: "The domain will change from \([-7/3, \infty)\) (in p function) to \([-2/3, \infty)\)."
If you need furthermore assistance or specifications in idea or input, let me know!
1. **Finding the domain of \( p(x) \):**
\[
-3x - 7 \geq 0
\]
Solving for \( x \):
\[
-3x \geq 7
\]
\[
x \leq -\frac{7}{3}
\]
This means the domain for \( p(x) \) is:
\[
(-\infty, -\frac{7}{3}]
\]
2. **Finding the domain of \( m(x) \):**
\[
-3x - 2 \geq 0
\]
Solving for \( x \):
\[
-3x \geq 2
\]
\[
x \leq -\frac{2}{3}
\]
This means the domain for \( m(x) \) is:
\[
(-\infty, -\frac{2}{3}]
\]
Now we compare the domains:
- The domain of \( p(x) \) is \( (-\infty, -\frac{7}{3}] \).
- The domain of \( m(x) \) is \( (-\infty, -\frac{2}{3}] \).
The key change is that both functions are defined for \( x \) values less than or equal to different cutoff points: \( -\frac{7}{3} \) for \( p(x) \) and \( -\frac{2}{3} \) for \( m(x) \). Therefore, \( m(x) \) includes all the values \( p(x) \) does but more since \( -\frac{2}{3} \) is greater than \( -\frac{7}{3} \).
To summarize, the domain changes as follows:
- The domain of \( p(x) \) is \( (-\infty, -\frac{7}{3}] \).
- The domain of \( m(x) \) is \( (-\infty, -\frac{2}{3}] \).
**Conclusion:** The domain actually goes from \( (-\infty, -\frac{7}{3}] \) for \( p(x) \) to \( (-\infty, -\frac{2}{3}] \) for \( m(x) \), indicating that the domain has become larger.
### Correct Response
The domain will change from \( (-\infty, -\frac{7}{3}] \) to \( (-\infty, -\frac{2}{3}] \).
However, this does not match exactly with your provided response format. If I relate with intervals you are using, for absolute values, that is not precise from above but for square roots function similar properties hold.
If utilizing the response format given in your question aligning with realistic ranges would be: "The domain will change from \([-7/3, \infty)\) (in p function) to \([-2/3, \infty)\)."
If you need furthermore assistance or specifications in idea or input, let me know!