Question

What is the domain and range of y=−233x−7−−−−−√+1
?(1 point)
Responses

domain: x≤73
, range: y≤−23
domain: x is less than or equal to 7 thirds, range: y is less than or equal to negative 2 thirds

domain: x≥73
, range: y≥1
domain: x is greater than or equal to 7 thirds, range: y is greater than or equal to 1

domain: x≥−73
, 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≥73
, range: y≤1

Answer 1

To determine the domain and range of the function \( y = -\sqrt{233x - 7} + 1 \), we need to consider the expression inside the square root and how it affects the output of the function.

### Domain:
The expression \( 233x - 7 \) must be non-negative because the square root of a negative number is undefined in the set of real numbers. Therefore, we set up the inequality:

\[
233x - 7 \geq 0
\]

Solving for \( x \):

\[
233x \geq 7
\]
\[
x \geq \frac{7}{233}
\]

So, the domain of the function is:

\[
\text{Domain: } x \geq \frac{7}{233}
\]

### Range:
Now we consider the range. The square root function \( \sqrt{233x - 7} \) will yield values from 0 to positive infinity (as \( x \) goes up), which means \( -\sqrt{233x - 7} \) will yield values from 0 to negative infinity. Adding 1 shifts the entire function upwards by 1. Therefore, the range will be:

\[
y = -\sqrt{233x - 7} + 1 \implies y \leq 1
\]

### Final Answer:
Putting it all together:
- **Domain:** \( x \geq \frac{7}{233} \)
- **Range:** \( y \leq 1 \)

Thus, the correct response is:
**domain: \( x \geq \frac{7}{233} \), range: \( y \leq 1 \)**.