I doubt there is a standard tool/technique for this kind of task. Nevertheless, there are some approaches. You would need at least one of the following strategies, according to ref. 1 (ch. 8):
dimension subsetting: selecting some of the dimensions to display.
dimension reduction: transforming the data into a lower-dimensional dataset.
dimension embedding: mapping dimensions to graphical attributes such as
color, size, and shape.
multiple displays: showing multiple plots at the same time with different
"views" of your data.
A common approach is to use a scatterplot matrix. This consists of a grid of scatterplots, with the grid having N² cells, where N is the number of dimensions, as the following figure.
Notice that a strategy does not (necessarily) exclude another one. In the jargon of visualization this "strategies" would fall in the following categories:
Regarding your specific problem, I would try to use one of the following:
- Multiple displays with contour maps where the value of the functions would correspond to a hyperslice. To get the value of the slice you would need to have an interactive setting, e.g., sliders to control each component. As shown in the following image (from 2).
- A single contour map where you pick two of the variables $x_i$, $x_j$ and aggregate the remaining ones. You could think of showing the maximum, minimum, or mean, for example.
Those are the main options that come to my mind right now. In any case, I would suggest to check chapter 8 of reference 1, chapters 11-13 of reference 2 and take a glance at reference 3.
Ward, Matthew O., Georges Grinstein, and Daniel Keim. Interactive data visualization: foundations, techniques, and applications. AK Peters/CRC Press, 2015.
Tamara Munzner. Visualization Analysis and Design. A K Peters Visualization Series, CRC Press, 2014.
Constantine, Paul G. Active subspaces: Emerging ideas for dimension reduction in parameter studies. Vol. 2. SIAM, 2015.