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When dealing with square root functions, understanding the **domain** is crucial. It tells us the set of possible x-values that the function can accept. The reason square root functions have a limited domain is that you cannot take the square root of a negative number without entering the realm of complex numbers.
For the given function, \(y = \sqrt{9x - 153} - 5\), the expression inside the square root, \(9x - 153\), must be non-negative, meaning it must be zero or positive.
This can be expressed mathematically as:

• \(9x - 153 \geq 0\)
• Solving this inequality, \(x \geq 17\).

So the domain is all real numbers \(x\) that are greater than or equal to 17, written as \([17, \infty)\).

The **range** is the set of all possible y-values. Since the output of \(\sqrt{9x - 153}\) starts at 0 (when \(x = 17\)) and increases as \(x\) increases, the smallest value of \(y\) is \(-5\). Therefore, the range of this function is \([-5, \infty)\).

The y-intercept is a special point where the graph of a function meets the y-axis. To find it, we typically substitute \(x = 0\) into the function to see what \(y\) would be.
However, for the function \(y = \sqrt{9x - 153} - 5\), placing \(x = 0\) doesn't provide a valid point because substituting this value results in the expression inside the square root being negative:

• \(9(0) - 153 = -153\)

As square roots of negative numbers aren't real under typical mathematical conventions, the function truly begins at \(x = 17\).
This means there is no y-intercept because the domain does not include \(x = 0\). The graph starts only from the point \((17, 0)\) and moves upward.

Understanding transformations helps in sketching the graph efficiently.
This function, \(y = \sqrt{9x - 153} - 5\), has undergone a couple of transformations from the basic square root function \(y = \sqrt{x}\).

1. Horizontal Shift

The expression \(9x - 153\) implies a horizontal shift. To find this shift, solve \(9x = 153\) gives \(x = 17\), indicating the graph shifts to the right by 17 units.

2. Vertical Shift

The "-5" outside the square root signifies a vertical shift downward by 5 units.
Combining these transformations, the basic square root curve \(y = \sqrt{x}\) now starts at the point \((17, 0)\) and then moves upwards, shifted down by 5 units so the starting y-value is at 0 on the graph. Keep these shifts in mind to graph transformations of the square root functions accurately.