# State the parent function, transformations, domain and range. 19610701771.jpg

State the parent function, transformations, domain and range.

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Step 1
Given, $$\displaystyle{f{{\left({x}\right)}}}={2}^{{{\left({x}-{3}\right)}}}+{2}$$
The graph of f(x) is as shown :

Step 2
Parent function:$$\displaystyle{2}^{{{x}}}$$
Transformation :
1.Right shift by 3
2.Vertical shift of +2
Domain, D:$$\displaystyle{\left(-\infty,\infty\right)}$$
$$\displaystyle\lim_{{{X}\rightarrow-\infty}}{f{{\left({x}\right)}}}=\lim_{{{x}\rightarrow-\infty}}{\left({2}^{{{x}-{3}}}+{2}\right)}={2}$$
$$\displaystyle\lim_{{{X}\rightarrow\infty}}{f{{\left({x}\right)}}}=\lim_{{{x}\rightarrow\infty}}{\left({2}^{{{x}-{3}}}+{2}\right)}=\infty$$
Range,R:$$\displaystyle{\left[{2},\infty\right)}$$