Question

the function $f(x)=-sqrt{x}$ is represented by the graph. what is the range of the given function? \\(\circ\\) \\(\\{y \mid \text{all real numbers}\\}\\) \\(\circ\\) \\(\\{y \mid y \leq 0\\}\\) \\(\circ\\) \\(\\{y \mid y \geq 0\\}\\)

Explanation:

1. Recall the range of the square root function: The function \( y = \sqrt{x} \) has a range of \( \{y | y \geq 0\} \) because the square root of a non - negative number \( x \) (since \( x\geq0 \) for \( \sqrt{x} \) to be real) is always non - negative.
2. Analyze the transformation: For the function \( f(x)=-\sqrt{x} \), we are taking the negative of \( \sqrt{x} \). If \( \sqrt{x}\geq0 \), then multiplying by - 1 will reverse the inequality. So, \( -\sqrt{x}\leq0 \). This means that the output values (the \( y \) - values) of the function \( f(x) = -\sqrt{x} \) are all real numbers less than or equal to 0. Also, from the graph, we can see that the curve starts at the origin (0,0) and goes downwards as \( x \) increases, which confirms that the \( y \) - values are non - positive.

Answer:

\(\{y|y\leq0\}\)