Practicing Success
Find the range of the following function. $f(x)=\frac{1}{x^2+2}$ |
$(-\frac{1}{2},2]$ $(\frac{1}{2},0]$ $(0,\frac{1}{2}]$ $(-\frac{1}{2},\frac{1}{2}]$ |
$(0,\frac{1}{2}]$ |
$f(x)=\frac{1}{x^2+2}$ Above function is not defined for those values of x for which $x^2+2=0$. But there are no real values of x for which $x^2 = -2$. So, we can take any real value of x. Therefore, domain is set R. For range let $y=\frac{1}{x^2+2}$ So, $x^2+2=\frac{1}{y}$ or $x^2=\frac{1}{y}-2=\frac{1-2y}{y}$ Now $x^2≥0$ for all real values of x. $∴\frac{1-2y}{y}≥0$ $⇒\frac{2y-1}{y}≤0⇒0<y≤1/2$ Therefore, range is $(0,\frac{1}{2}]$ |