It is desired that the domain and range of the function should, respectively, be
[tex]\begin{gathered} \text{Domain}=(-\infty,\infty) \\ \text{Range}=(-\infty,2) \end{gathered}[/tex]Observe the given choices of function.
It is evident that all the functions are exponential functions, so their domain must be the set of all real numbers,
[tex](-\infty,\infty)[/tex]Now, we have to check the range of each of the 4 given functions.
Option A:
The function is given as,
[tex]y=-(0.25)^x-2[/tex]Consider the following,
[tex]\begin{gathered} x\rightarrow\infty\Rightarrow-(0.25)^x\rightarrow0\Rightarrow-(0.25)^x-2\rightarrow-2\Rightarrow y\rightarrow-2 \\ x\rightarrow-\infty\Rightarrow-(0.25)^x\rightarrow-\infty\Rightarrow-(0.25)^x-2\rightarrow-\infty\Rightarrow y\rightarrow-\infty \end{gathered}[/tex]Thus, we see that the range of the function is,
[tex]\text{Range}=(-\infty,-2)[/tex]Since this does not match with the desired range. This is not a correct choice.
Option B:
The function is given as,
[tex]y=-(3)^x-2[/tex]Consider the following,
[tex]\begin{gathered} x\rightarrow\infty\Rightarrow-(3)^x\rightarrow-\infty\Rightarrow-(3)^x-2\rightarrow-\infty\Rightarrow y\rightarrow-\infty \\ x\rightarrow-\infty\Rightarrow-(3)^x\rightarrow0\Rightarrow-(3)^x-2\rightarrow-2\Rightarrow y\rightarrow-2 \end{gathered}[/tex]Thus, we see that the range of the function is,
[tex]\text{Range}=(-\infty,-2)[/tex]Since this does not match with the desired range. This is not a correct choice.
Option C:
The function is given as,
[tex]y=-(3)^x+2[/tex]Consider the following,
[tex]\begin{gathered} x\rightarrow\infty\Rightarrow-(3)^x\rightarrow-\infty\Rightarrow-(3)^x+2\rightarrow-\infty\Rightarrow y\rightarrow-\infty \\ x\rightarrow-\infty\Rightarrow-(3)^x\rightarrow0\Rightarrow-(3)^x+2\rightarrow2\Rightarrow y\rightarrow2 \end{gathered}[/tex]Thus, we see that the range of the function is,
[tex]\text{Range}=(-\infty,2)[/tex]Since this exactly matches with the desired range. This is a correct choice.
Option D:
The function is given as,
[tex]y=-(0.25)^x+2[/tex]Consider the following,
[tex]\begin{gathered} x\rightarrow\infty\Rightarrow-(0.25)^x\rightarrow0\Rightarrow-(0.25)^x+2\rightarrow2\Rightarrow y\rightarrow2 \\ x\rightarrow-\infty\Rightarrow-(0.25)^x\rightarrow-\infty\Rightarrow-(0.25)^x-2\rightarrow-\infty\Rightarrow y\rightarrow-\infty \end{gathered}[/tex]Thus, we see that the range of the function is,
[tex]\text{Range}=(-\infty,2)[/tex]Since this exactly matches with the desired range. This is also a correct choice.
Thus, the we see that the functions in option C and D possess the desired domain and range.
Therefore, option C and option D are t
