Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \frac{2x+7}{x-2}$,

Question

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \frac{2x+7}{x-2}$, $x\in \mathbb{R},$$x\ne 2$, $\mathrm{g}:x\mapsto {{\left( x-1 \right)}^{2}}-2$, $x\in \mathbb{R}$, $0 < x < 3$ Show that the composite function $\mathrm{fg}$ exists.[2] Define $\mathrm{fg}$ in a similar form and find its range.[3]

Accepted Answer

(i) Since ${{\mathrm{R}}_{\mathrm{g}}}\subseteq {{\mathrm{D}}_{\mathrm{f}}}$, $\mathrm{fg}$ exists. (ii) $\mathrm{fg}:x\mapsto \frac{2{{x}^{2}}-4x+5}{{{x}^{2}}+2x-3}$ $x\in \mathbb{R},$ $0<x<3$. (Also accept $\mathrm{f}\left( \mathrm{g}\left( x \right) \right)=2+\frac{11}{{{x}^{2}}-2x-3}=\frac{2{{\left( x-1 \right)}^{2}}+3}{{{\left( x-1 \right)}^{2}}-4}$)

2023 TMJC Promo Q2

2023 TMJC Promo Q2

Video Solution:

Solution:

Solution locked