Given:
The range of a function is [tex]y<3[/tex].
To find:
The function for the given range from the given options.
Solution:
In option A, the given function is:
[tex]y=3(2)^x[/tex]
Here, [tex](2)^x[/tex] is always greater than 0. So, [tex]3(2)^x[/tex] is also greater than 0, i.e., [tex]y>0[/tex].
In option B, the given function is:
[tex]y=2(3)^x[/tex]
Here, [tex](3)^x[/tex] is always greater than 0. So, [tex]2(3)^x[/tex] is also greater than 0, i.e., [tex]y>0[/tex].
In option C, the given function is:
[tex]y=-(2)^x+3[/tex]
Here,
[tex](2)^x>0[/tex]
[tex]-(2)^x<0[/tex]
[tex]-(2)^x+3<0+3[/tex]
[tex]y<3[/tex]
The range of this function is [tex]y<3[/tex]. So, option C is correct.
In option D, the given function is:
[tex]y=(2)^x-3[/tex]
Here,
[tex](2)^x>0[/tex]
[tex](2)^x-3<0-3[/tex]
[tex]y<-3[/tex]
The range of this function is [tex]y<-3[/tex]
Therefore, the correct option is only C.
