Let be the rational function g_a(x)= x+a x-2

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

Let

g_a:\,\mathbb R\setminus\{2\}\to\mathbb R\setminus\{1\}

be the rational function

g_a(x)=\frac{x+a}{x-2},\quad a\in\mathbb R.

Answer the following. (i) Determine the range of

g_a

in terms of

a

. (ii) Find all real values of

a

for which

g_a

is onto

\mathbb R\setminus\{1\}

. (iii) For each value of

a

obtained in part (ii), obtain

g_a^{-1}(x)

and specify its domain and range.