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

Answer 1

To find the domain and range of the function \( y = -\frac{2}{3}\sqrt{3x - 7} + 1 \), we need to analyze the expression step by step.

### Domain
The expression \(\sqrt{3x - 7}\) requires that the argument of the square root be non-negative. Thus, we need to solve:

\[
3x - 7 \geq 0
\]

Solving this inequality:

\[
3x \geq 7
\]
\[
x \geq \frac{7}{3}
\]

Therefore, the domain of the function is:

\[
x \geq \frac{7}{3}
\]

### Range
Next, we find the range of \( y \). The term \(\sqrt{3x - 7}\) will reach a minimum value of 0 when \( x = \frac{7}{3} \) and will increase as \( x \) increases.

Substituting \( x = \frac{7}{3} \):

\[
y = -\frac{2}{3}\sqrt{3(\frac{7}{3}) - 7} + 1 = -\frac{2}{3}(0) + 1 = 1
\]

As \( x \) increases, \(\sqrt{3x - 7}\) will increase, and thus \( y \) will decrease (because of the negative sign in front of the \(-\frac{2}{3}\)):

As \( \sqrt{3x - 7} \to \infty \), \( y \to -\infty \).

Thus, the maximum value of \( y \) is 1 (when \( x = \frac{7}{3} \)) and it decreases indefinitely. Therefore, the range of \( y \) is:

\[
y \leq 1
\]

### Conclusion
The domain and range of the function \( y = -\frac{2}{3}\sqrt{3x - 7} + 1 \) are as follows:

- **Domain:** \( x \geq \frac{7}{3} \)
- **Range:** \( y \leq 1 \)

So the correct answer from the options given is:

**domain: x ≥ 7/3, range: y ≤ 1**.