Question

Select all that are always true of a function that has no real roots: (2 points)
Responses

the graph does not touch the x-axis
the graph does not touch the x-axis

the graph does not touch the y-axis
the graph does not touch the y-axis

the solutions are irrational numbers
the solutions are irrational numbers

the solutions are complex numbers

Answer 1

For a function that has no real roots, the following statements are always true:

1. **The graph does not touch the x-axis.** - This indicates that the function does not have any real solutions (roots).

2. **The solutions are complex numbers.** - Since the function has no real roots, any roots it may have must be complex.

The other two statements are not generally true:
- The graph does touch the y-axis since the y-axis is where \( x=0 \); a function can still touch the y-axis while having no real roots.
- The solutions can be complex but are not necessarily irrational; they could also be rational complex numbers.

So, the correct selections for a function with no real roots are:
- The graph does not touch the x-axis.
- The solutions are complex numbers.