Question
How does the domain change from p(x)=−3x−7−−−−√+6
to m(x)=−3x−2−−−−√+6
?(1 point)
Responses
The domain will not change.
The domain will not change.
The domain will change from [2,∞)
to [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,∞)
to [2,∞)
.
The domain will change from left bracket 7 comma infinity right parenthesis to left bracket 2 comma infinity right parenthesis .
The range will change from [7,∞)
to [2,∞)
.
to m(x)=−3x−2−−−−√+6
?(1 point)
Responses
The domain will not change.
The domain will not change.
The domain will change from [2,∞)
to [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,∞)
to [2,∞)
.
The domain will change from left bracket 7 comma infinity right parenthesis to left bracket 2 comma infinity right parenthesis .
The range will change from [7,∞)
to [2,∞)
.
Answers
Answered by
GPT-4o mini
To understand how the domain of the functions \( p(x) = -\sqrt{-3x - 7} + 6 \) and \( m(x) = -\sqrt{-3x - 2} + 6 \) changes, we need to analyze the inside of the square root for each function, since the expression under the square root must be non-negative for the function to have real values.
1. **For \( p(x) = -\sqrt{-3x - 7} + 6 \)**:
- The expression under the square root is \( -3x - 7 \).
- Set the expression \( -3x - 7 \geq 0 \):
\[
-3x - 7 \geq 0 \\
-3x \geq 7 \\
x \leq -\frac{7}{3}
\]
- So the domain of \( p(x) \) is \( (-\infty, -\frac{7}{3}] \).
2. **For \( m(x) = -\sqrt{-3x - 2} + 6 \)**:
- The expression under the square root is \( -3x - 2 \).
- Set the expression \( -3x - 2 \geq 0 \):
\[
-3x - 2 \geq 0 \\
-3x \geq 2 \\
x \leq -\frac{2}{3}
\]
- So the domain of \( m(x) \) is \( (-\infty, -\frac{2}{3}] \).
Now comparing the domains:
- The domain of \( p(x) \) is \( (-\infty, -\frac{7}{3}] \)
- The domain of \( m(x) \) is \( (-\infty, -\frac{2}{3}] \)
Since \(-\frac{2}{3}\) is greater than \(-\frac{7}{3}\), we can conclude:
**The domain changes from \( (-\infty, -\frac{7}{3}] \) to \( (-\infty, -\frac{2}{3}] \).**
However, since these are potential options from your responses, if you have them available for evaluation (like in brackets), recheck for matching intervals provided for the exact transition. Based on your response options, the correct choice seems different, but based on the analysis, the domains indeed shift upward in value range.
1. **For \( p(x) = -\sqrt{-3x - 7} + 6 \)**:
- The expression under the square root is \( -3x - 7 \).
- Set the expression \( -3x - 7 \geq 0 \):
\[
-3x - 7 \geq 0 \\
-3x \geq 7 \\
x \leq -\frac{7}{3}
\]
- So the domain of \( p(x) \) is \( (-\infty, -\frac{7}{3}] \).
2. **For \( m(x) = -\sqrt{-3x - 2} + 6 \)**:
- The expression under the square root is \( -3x - 2 \).
- Set the expression \( -3x - 2 \geq 0 \):
\[
-3x - 2 \geq 0 \\
-3x \geq 2 \\
x \leq -\frac{2}{3}
\]
- So the domain of \( m(x) \) is \( (-\infty, -\frac{2}{3}] \).
Now comparing the domains:
- The domain of \( p(x) \) is \( (-\infty, -\frac{7}{3}] \)
- The domain of \( m(x) \) is \( (-\infty, -\frac{2}{3}] \)
Since \(-\frac{2}{3}\) is greater than \(-\frac{7}{3}\), we can conclude:
**The domain changes from \( (-\infty, -\frac{7}{3}] \) to \( (-\infty, -\frac{2}{3}] \).**
However, since these are potential options from your responses, if you have them available for evaluation (like in brackets), recheck for matching intervals provided for the exact transition. Based on your response options, the correct choice seems different, but based on the analysis, the domains indeed shift upward in value range.