For a function that has no real roots, here are the true statements:
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The solutions are complex numbers - This is always true if the function has no real roots. Complex roots exist when a polynomial does not intersect the x-axis.
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The graph does not touch the x-axis - This is also true for a function with no real roots. If it has no real roots, it does not cross or touch the x-axis.
The other statements about solutions being irrational and the graph not touching the y-axis are not always true. The graph can still touch the y-axis for functions with no real roots, and the roots can be complex (which may not necessarily be irrational).
Therefore, the statements that are always true are:
- The solutions are complex numbers.
- The graph does not touch the x-axis.