\(ex.\,\,a\)The domain of a function is what can go into a function. The range is what can come out.
The domain is found by noting restrictions on the function, while the range is found by noting restrictions on the same function rearranged in terms of the other variable type.
The restrictions are NOT in the domain or range, respectively
\(What\,\, is\,\, the\,\, domain\,\, and\)\(range\,\, of\,\, y = \dfrac{1}{2 - x}? \)
\(The \,\,domain \,\,is \,\, (-\infty, 2) \cup (2, \infty)\)
\(The\,\,reason\,\,being\,\,that\,\,any\)\(real\,\,number \,\,but\,\,2\,\,will \,\,go\,\,into\,\,x.\)
\(The\,\,range\,\,is\,\, (-\infty, 0) \cup (0, \infty)\) \(because \,\,the\,\,restriction\,\,on\,\,\)\(the\,\,rearranged\,\,equation.\)
\(y = \dfrac{1}{2 - x}\)
\((2 - x)y = (2 - x)\dfrac{1}{2 - x}\)
\(2y - xy = 1\)
\(\dfrac{2y}{y} - \dfrac{xy}{y} = \dfrac{1}{y}\)
\(2 - x = \dfrac{1}{y}\)
\(2 - 2 - x = \dfrac{1}{y} - 2\)
\(-x = \dfrac{1}{y} - 2\)
\(-x(-1) = \dfrac{1}{y}(-1) - 2(-1)\)
\(x = -\dfrac{1}{y} + 2\)
