Question

Let $R$ be the set of real numbers and the functions $f:R\to R$ and $g:R\to R$ be defined by $f\left(x\right)=x^2+2x-3$ and $g\left(x\right)=x+1$. Then the value of $x$ for which $f\left(g\right(x\left)\right)=g\left(f\right(x\left)\right)$ is

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is A $-1$

According to the question,

$f\left(g\right(x\left)\right)=g\left(f\right(x\left)\right)$
$\Rightarrow f(x+1)=g(x^2+2x-3)$

$\Rightarrow (x+1{)}^2+2(x+1)-3=x^2+2x-3+1$

$\Rightarrow x^2+1+2x+2x+2-3=x^2+2x-2$

$\Rightarrow x^2+4x=x^2+2x-2$
$\Rightarrow x^2+4x-x^2-2x+2=0$
$\Rightarrow 2x+2=0$

$\Rightarrow 2x=-2$

$\Rightarrow x=-1$