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

Answer 1

Answer: (A), (B)

The correct options are
A $R_f=[{\log }_e\frac{11}{3},\infty )$
B $D_f=R$

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