Question

In Exercises 13-20, use a grapher to (a) identify the domain and range and (b) draw the graph of the function. $$y=2 \sqrt{3-x}$$

Short answer

The domain of the function is \( (-\infty, 3] \) and the range is \( [0, +\infty) \). The graph starts from \( x = 3 \), is at \( y = 0 \) at this point, and increases as \( x \) decreases. It exhibits reflection across the y-axis.

Step-by-step solution

Find the domain

The domain of a function is the set of all possible input values (i.e., the \( x \) variable) which will create a valid output. In this function, as it involves the square root of \( 3-x \), the expression under the square root (the radicand) must be greater or equal to zero (because in the real number system, the square root of a negative number doesn't exist). So, we have: \( 3 - x \geq 0 \). Solving this inequality for \( x \), we find \( x \leq 3 \). So, our domain is \( (-\infty, 3] \), meaning it can be any value less than or equal to 3.

Find the range

The range of a function is the set of all possible output values given the domain. The square root function will output only nonnegative values, i.e., values that are greater or equal to zero. So the range of the function is \( [0, +\infty) \).

Draw the graph

Now, graph the function \( y = 2\sqrt{3-x} \) starting from \( x \) equals 3 and moving backward, since the valid \( x \) values are \( 3 \) or less. Remember that the function will output zero when \( x = 3 \) and increases as \( x \) becomes lower. As this is a negative square root function, the graph will appear in the 2nd quadrant, reflecting across the y-axis.