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the function is:
f[x_] = a^x + Sqrt[a^x + 2]
In[357]:= Clear["Global`*"]
f[x_] = a^x + Sqrt[a^x + 2]
FunctionRange[{f[x], a > 0 && a != 1}, x, y]
Out[358]= a^x + Sqrt[2 + a^x]
During evaluation of In[357]:= FunctionRange::nmet: Unable to find the range with the available methods.
Out[359]= FunctionRange[{a^x + Sqrt[2 + a^x], a > 0 && a != 1}, x, y]
```
Clear["Global`*"]
f[x_] = a^x + Sqrt[a^x + 2]
FunctionRange[{f[x], a > 0 && a != 1}, x, y, Method -> {"Reduced" -> True}]
