Question
What is the domain and range of y=−2/3√3x - 7 +1
?(1 point)
Responses
domain: x≤7/3
, range: y≤−2/3
domain: x is less than or equal to 7 thirds, range: y is less than or equal to negative 2 thirds
domain: x≥7/3
, range: y≤1
domain: x is greater than or equal to 7 thirds, range: y is less than or equal to 1
domain: x≥−7/3
, range: y≤−1
domain: x is greater than or equal to negative 7 thirds, range: y is less than or equal to negative 1
domain: x≥7/3
, range: y≥1
?(1 point)
Responses
domain: x≤7/3
, range: y≤−2/3
domain: x is less than or equal to 7 thirds, range: y is less than or equal to negative 2 thirds
domain: x≥7/3
, range: y≤1
domain: x is greater than or equal to 7 thirds, range: y is less than or equal to 1
domain: x≥−7/3
, range: y≤−1
domain: x is greater than or equal to negative 7 thirds, range: y is less than or equal to negative 1
domain: x≥7/3
, range: y≥1
Answers
GPT-4o mini
To find the domain and range of the function \( y = -\frac{2}{3}\sqrt{3x} - 7 + 1 \), let's simplify the equation first:
\[
y = -\frac{2}{3}\sqrt{3x} - 6
\]
### Domain:
The expression \(\sqrt{3x}\) is defined when \(3x \geq 0\), which gives:
\[
x \geq 0
\]
Therefore, the domain is:
\[
\text{Domain: } x \geq 0
\]
### Range:
The function \(y = -\frac{2}{3}\sqrt{3x} - 6\) involves a square root, and since the square root is always non-negative, the term \(-\frac{2}{3}\sqrt{3x}\) will be non-positive, meaning it will always decrease from \(0\) (when \(x = 0\)) downwards.
When \(x = 0\):
\[
y = -\frac{2}{3}\sqrt{3(0)} - 6 = -6
\]
As \(x\) increases, \(\sqrt{3x}\) increases, leading \(y\) to decrease without bound. Therefore, the lowest value of \(y\) is \(-6\), with no upper limit (since \(y\) can be less than \(-6\) as \(x\) increases).
So, the range is:
\[
\text{Range: } y \leq -6
\]
### Conclusion:
Putting it together, the correct interpretation of the domain and range is:
- Domain: \(x \geq 0\)
- Range: \(y \leq -6\)
Since the provided options do not exactly match a typical calculation of domain and range, there's no suitable match in the listed responses. The closest appropriate interpretation is that the function's outputs can decrease indefinitely, starting from \(-6\) at \(x=0\) and lower as \(x\) increases.
**If choosing the closest answer:**
The appropriate response based on the structure of the question would be:
- domain: \(x \geq 0\), range: \(y \text{ approaches } -6 \text{ as } x \text{ increases}.\)
But this isn't strictly listed among the multiple-choice options provided, so it suggests a misunderstanding in their context. Ensure there's no other option to re-check that better matches.
\[
y = -\frac{2}{3}\sqrt{3x} - 6
\]
### Domain:
The expression \(\sqrt{3x}\) is defined when \(3x \geq 0\), which gives:
\[
x \geq 0
\]
Therefore, the domain is:
\[
\text{Domain: } x \geq 0
\]
### Range:
The function \(y = -\frac{2}{3}\sqrt{3x} - 6\) involves a square root, and since the square root is always non-negative, the term \(-\frac{2}{3}\sqrt{3x}\) will be non-positive, meaning it will always decrease from \(0\) (when \(x = 0\)) downwards.
When \(x = 0\):
\[
y = -\frac{2}{3}\sqrt{3(0)} - 6 = -6
\]
As \(x\) increases, \(\sqrt{3x}\) increases, leading \(y\) to decrease without bound. Therefore, the lowest value of \(y\) is \(-6\), with no upper limit (since \(y\) can be less than \(-6\) as \(x\) increases).
So, the range is:
\[
\text{Range: } y \leq -6
\]
### Conclusion:
Putting it together, the correct interpretation of the domain and range is:
- Domain: \(x \geq 0\)
- Range: \(y \leq -6\)
Since the provided options do not exactly match a typical calculation of domain and range, there's no suitable match in the listed responses. The closest appropriate interpretation is that the function's outputs can decrease indefinitely, starting from \(-6\) at \(x=0\) and lower as \(x\) increases.
**If choosing the closest answer:**
The appropriate response based on the structure of the question would be:
- domain: \(x \geq 0\), range: \(y \text{ approaches } -6 \text{ as } x \text{ increases}.\)
But this isn't strictly listed among the multiple-choice options provided, so it suggests a misunderstanding in their context. Ensure there's no other option to re-check that better matches.