91Ó°ÊÓ

The domain of a function refers to the set of all possible input values (usually represented as \( x \)) for which the function is defined.
For the function \( y = \sqrt{x - 4} \), the expression inside the square root, \( x - 4 \), must be non-negative because the square root of a negative number is not defined in the set of real numbers.
To find the domain, solve the inequality:

• \( x - 4 \geq 0 \)
• Adding 4 to both sides gives \( x \geq 4 \)

Thus, the domain of the function is all real numbers \( x \) such that \( x \geq 4 \), which is written in interval notation as \([4, \infty)\).
This means the function exists and has real values starting from \( x = 4 \) onwards.

Intercepts are the points where a graph crosses the axes. There are two types:

• X-intercept: Where the graph crosses the x-axis (i.e., \( y = 0 \))
• Y-intercept: Where the graph crosses the y-axis (i.e., \( x = 0 \))

For the function \( y = \sqrt{x - 4} \):

• X-intercept: Set \( y = 0 \). Solving \( \sqrt{x - 4} = 0 \) gives \( x = 4 \). Thus, the x-intercept is at the point \((4, 0)\).
• Y-intercept: Set \( x = 0 \). Then \( y = \sqrt{0 - 4} \) results in an undefined value since the square root of a negative number is not a real number. Hence, there is no y-intercept for this function.

These intercepts indicate where the graph will be positioned in relation to the coordinate axes.

Graphing functions is a vital skill in understanding how equations behave visually. To sketch a graph of \( y = \sqrt{x - 4} \), begin by plotting the x-intercept, which we found to be \( (4,0) \).
This point is crucial, as it marks the beginning of the function's graph due to its domain being \( [4, \infty) \).
From the x-intercept, as you choose values of \( x \) greater than 4, compute additional points such as

• \( (5,1) \) because \( \sqrt{5-4} = 1 \)
• \( (8,2) \) since \( \sqrt{8-4} = 2 \)

Plot these points and connect them smoothly with a curve that steadily rises to the right, illustrating the graph's half-parabola shape.
The end behavior of the graph is also important; as \( x \) tends to infinity, \( y = \sqrt{x-4} \) also increases infinitely.
This insight helps anticipate how the graph will behave outside the plotted points. By understanding these key aspects, you can accurately depict the function on a graph.