The range of the function \( f(x)=5-\sqrt{x-3} \) is \[ \begin{array}{r}{[3,+\infty) \bigcirc} \\ \\ (-\infty, 3] \bigcirc \\ \\ {[5,+\infty, 5]}\end{array} \]

To find the range of the function \( f(x) = 5 - \sqrt{x-3} \), we first note that the square root function \( \sqrt{x-3} \) is defined for \( x \geq 3 \). As \( x \) approaches 3, \( \sqrt{x-3} \) approaches 0, making \( f(3) = 5 - 0 = 5 \). As \( x \) increases, \( \sqrt{x-3} \) increases without bound, causing \( f(x) \) to decrease without bound. Therefore, \( f(x) \) can take any value from \( 5 \) down to negative infinity. Thus, the range of \( f(x) \) is \( (-\infty, 5] \). When looking at the problem visually or using a graphing tool, you can see how the function behaves as you plug in different values for \( x \). It’s quite fascinating to watch the tip of the curve at \( (3, 5) \) descend as you head towards larger values of \( x \). Understanding these concepts not only helps solidify your mathematical skills but also encourages a deeper appreciation for the elegance of functions!