Question

Which of the following are true for the function $f\left(x\right)={\log }_e(3x^2-4x+5)$ (where $D_f,R_f$ represents domain and range of function $f\left(x\right)$ reapectively)

• A. $R_f=R$
• B. $D_f=R$
• C. $D_f=[0,\infty )$
• D. $R_f=[{\log }_e\frac{11}{3},\infty )$

Answer 1

Answer: (B), (D)

$R_f=[{\log }_e\frac{11}{3},\infty )$

$f\left(x\right)$ is defined if $3x^2-4x+5>0$
$\Rightarrow 3[x^2-\frac{4}{3}x+\frac{5}{3}]>0\Rightarrow 3[{(x-\frac{2}{3})}^2+\frac{11}{9}]>0$
Which is true for all real $x$
Domain of $f\left(x\right)=(-\infty ,\infty )$

We can clearly see $3x^2-4x+5=3[{(x-\frac{2}{3})}^2+\frac{11}{9}]$
So, the quadratic expression is always $\ge \frac{11}{3}.$
$\Rightarrow$ Range of the quadratic expression is $[\frac{11}{3},\infty )$
$\therefore {\log }_e(3x^2-4x+5)\in [{\log }_e\frac{11}{3},\infty )$
Range of $f\left(x\right)=[{\log }_e\frac{11}{3},\infty )$