The graphs of four radical functions are shown in the image.
The radicals in the function rules are either square roots or cube roots. Thus, the parent function of each is either the square root of x, or the cube root of x. Let's start with Graph I.
From the image, we can see that the function corresponding to Graph I isn't defined for all real numbers. Thus, it must be one of the square root functions. Comparing it to the graph of can give us more information about its function rule.
Graph II is very similar to Graph I, so let's compare them.
Here, II looks to be a reflection of I in the x-axis. If this is the case, one of the options must be equal to -h(x).
With only one rule remaining, Graph IV must correspond to t. To confirm this, t can be viewed as a translation of 1 unit to the left and 0.3 units downward. This can also be seen in its graph.
The x-values and the function values can now be plotted as points (x,f(x)) in a coordinate plane. We'll connect the points with a smooth curve. Note that the function is not defined for x<0.
The points can now be plotted in the same coordinate plane as f, again connecting the points with a smooth curve.
Furthermore, the second transformation is a reflection in the x-axis.
Therefore, f has undergone both a horizontal translation and a reflection in the x-axis to become g.