Question

Find the domain and range of the following functions:

1) $\frac{1}{x-2}$

2) $\frac{x}{x+5}$

3) $\frac{1}{2+x^2}$

4) ${}^{7-x}{P}_{x-3}$

Answer 1

find the domain and range

$A\frac{1}{x-2}$

let $f\left(x\right)=\frac{1}{x-2}.$ here $x-2\ne 0$

$\Rightarrow \ne 2$

domain $=R-<2>$

$y=\frac{1}{n-2}\Rightarrow (n-2)y=1$

$\Rightarrow ny-2y=1\Rightarrow ny=1+2y$

$x=\frac{1+2y}{y},y\ne 0$

(B) $Range=R-<0>$

$y=\frac{x}{x+5}\Rightarrow x+5\ne 0\Rightarrow n\ne -5$ domain $=R-<5>$

$xy+5y=n\Rightarrow x(y-1)=-5y$

$\left(c\right)Range=R-<1>$ $n=\frac{5y}{1-y}\Rightarrow 1-y\ne 0$

$y=\frac{1}{2+x^2}\Rightarrow domain=R$ $y\ne 1$

$2y+x^{2}y=1\Rightarrow x^2=\frac{1-2y}{y}\Rightarrow x=\pm \sqrt{\frac{1-2y}{y}}$

$y\ne 0,1-2y⩾0\Rightarrow 2y\le 1$

$\Rightarrow y\le 1/2$ Range $=(0,1/2)$

$(D{)}^{7-x}{P}_{x-3}\Rightarrow$ Here $7-x>x-3\Rightarrow 7+3>2x\Rightarrow x<5$

${}^3{P}_1=\frac{L3}{L2}=3$ Ans so $x=4$ & $x>3$