Question

Find the domain and the range of the function. Then sketch the graph of the function. $$y=x \sqrt{8 x}$$

Short answer

The domain of the function \(y = x \sqrt{8x}\) is \([0, +\infty)\) and the range is also \([0, +\infty)\). The graph of the function starts at the origin, growing but slowing down as x increases and it stays in the first quadrant.

Step-by-step solution

Finding the Domain

Determine the domain of the function \(y = x \sqrt{8x}\). Recall that square roots are undefined for negative real numbers. Thus, \(x \sqrt{8x}\) is defined when \(x \geq 0\). So, the domain of this function is \([0, +\infty)\).

Finding the Range

For every x value in the domain, we need to find the corresponding output i.e. y value. The value inside the square root is always positive, and when this is multiplied by x (which is non-negative in our domain), the result will also be non-negative. Therefore, the range is also \([0, +\infty)\).

Sketching the Graph

The graph will start at (0,0) because when \(x = 0\), then \(y = 0\). As \(x\) increases, \(y = x \sqrt{8x}\) also increases. The graph will not intersect or touch the negative x and y axis because the domain and range are both \([0, +\infty)\). Thus, the graph will be in the first quadrant starting from the origin, growing but slowing down as x increases.