Question

comparing exponential functions based on tables
consider the function ( f(x) = 3^x ) and a translation of ( f(x) ) named ( g(x) ). janeen created a table for select values of ( x ). her table is shown below.

( x )	0	1	2	3
( g(x) )	3	9	27	81

what can be concluded about the relationship between the two functions? check all that apply.

• the functions have the same base.
• the functions have the same range.
• the functions have the same exponent.
• the functions have the same domain.
• ( g(x) ) is a translation left 1 unit.
• ( g(x) ) is a translation right 2 units.
• ( g(x) ) is a translation up 2 units.

Explanation:

• Same base: \(f(x) = 3^x\) and \(g(x) = 3^x + 2\) both have base 3.
• Translation up 2 units: For each \(x\), \(g(x) = f(x) + 2\) (e.g., \(g(0)=3=1+2=f(0)+2\), \(g(1)=9=3+2=f(1)+2\)), which is a vertical shift up by 2.

Other options are incorrect: the functions don’t have the same exponent/range/domain, and \(g(x)\) isn’t a horizontal translation.

Answer:

The functions have the same base., g(x) is a translation up 2 units.