y = 4^{x - 5}+3\nthe domain of this function is\n\nthe range of this function is\n\ndone

Explanation:

Step1: Find the domain

For an exponential function (y = a^{x}) (in this case (a = 4)), the exponent (x-5) can take any real - valued input. The domain of (y = 4^{x - 5}+3) is all real numbers. In interval notation, for any function (y = f(x)) where (f(x)) is defined for all (x\in(-\infty,\infty)), the domain is ((-\infty,\infty)).

Step2: Find the range

We know that for the exponential function (y = 4^{u}), where (u=x - 5), the range of (y = 4^{u}) is ((0,\infty)) since (4^{u}>0) for all real (u). For the function (y = 4^{x - 5}+3), we use the transformation (y = f(x)+k) (where (f(x)=4^{x - 5}) and (k = 3)). If (y=f(x)) has a range of ((0,\infty)), then (y=f(x)+3) has a range of ((0 + 3,\infty+3))

Answer:

• Domain: ((-\infty,\infty))
• Range: ((3,\infty))