91Ó°ÊÓ

Let's start by understanding the domain and range of a function. The **domain** of a function is the set of all possible input values (x-values) that the function can accept without causing any issues, like dividing by zero or taking the square root of a negative number.

In the given problem, we are working with the function: \( y = -13 + \sqrt{3x} \).
To find the domain, we need the expression inside the square root, \(3x\), to be non-negative:
\[ 3x \geq 0 \]
Solving this inequality for x, we get:
\[ x \geq 0 \]
Therefore, the domain of the function is **x** greater than or equal to 0.

Next, the **range** of a function is the set of all possible output values (y-values). For the given function, the minimum value occurs when \sqrt{3x}\ equals 0, giving us
\[ y = -13 + 0 = -13 \]
As **x** increases, \sqrt{3x}\ also increases, meaning there is no upper limit to **y**. Hence, the range of the function is **y** greater than or equal to -13.

Horizontal translation involves shifting the function left or right on the coordinate plane. However, the provided exercise does not include a horizontal translation. But let's understand it using related concepts.

Consider the base function \( y = \sqrt{x} \. If we replace **x** with \((x - h)\), this shifts the graph horizontally. If **h** is positive, the function moves **right**; if negative, the function moves **left**.

Comparing this to the given problem, the function \( y = -13 + \sqrt{3x} \) involves modifying only the coefficient inside the square root, which impacts the graph's shape but doesn't represent a translation. Therefore, there is no horizontal translation in this problem. The term \( 3x \) inside the square root causes horizontal compression by a factor of \sqrt{3} \).

Vertical translation affects the function by shifting it up or down on the coordinate plane. This translation is more straightforward compared to horizontal translation.

For the given base function \ y = \sqrt{x} \ adding |b| shifts it vertically.
In this exercise, our function \( y = 4 - \sqrt{3x} \) has been translated vertically by 9 units up:
\[ y_{translated} = 13 - \sqrt{3x} \]
This means each y-value of the function has increased by 9 units.
The vertical translation ensures that for every x-value, the value of y changes accordingly. The image function then gets reflected over the x-axis by multiplying by -1. So we have:
\[ y = -(13-\sqrt{3x})\]
Until we resolve to:
\[ y = -13 + \sqrt{3x}\]
So in the final function, we have effectively shifted it down by 13 units. And post-reflection the entire function shifts down and inverts.

Reflection is a transformation that flips a function over a specific axis. Reflections can be performed over either the x-axis or the y-axis.

In this exercise, the function is reflected over the x-axis. To reflect a function over the x-axis, you multiply the function by -1.

Starting from our vertically translated function:
\[ y = 13 - \sqrt{3x} \]
Multiply the entire function by -1:
\[ y_{reflected} = -(13 - \sqrt{3x}) = -13 + \sqrt{3x} \]
This flips the graph of the function upside down. Hence, all positive y-values become negative, and all negative y-values become positive (if any existed).

Thus, reflecting over the x-axis is like flipping the graph; the points stay horizontally aligned but switch y-coordinates from top to bottom or vice versa.