Is my fathers social security income of 9,228.00 considered taxable income?

It could be. If that is all the income he has, then no, it is not gross income, nor is it taxable.

You can claim a child, relative, friend, fiancé (etc.) as a dependent on your 2020 taxes as long as they meet all of the following requirements :

• You provided more than half of their financial support. More info
• They made less than $4,300 in gross income during 2020 unless they are a qualifying child.
• They live with you or they are related to you. (Your relative must live at your residence all year or be on the list of “relatives who do not live with you” in Publication 501.) 
• They are a U.S. citizen, resident alien, national, or a Canadian or Mexican resident.
• They aren't (or won't be) claimed as a dependent by someone else.
• They aren’t filing a joint return with their spouse.
• You are not being claimed as a dependent on someone else's return