Question

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

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

Answer 1

Answer: (A), (C)

$D_f=R$

$f\left(x\right)=3x^2-4x+5$ is defined for all real numbers.
Hence, domain of $f\left(x\right)=(-\infty ,\infty )$ i.e. $D_f:x\in R$

$R_f:y\in [-\frac{D}{4a},\infty )$
$a=3,b=-4,c=5$
$-\frac{D}{4a}=-\frac{(-4{)}^2-4\times 3\times 5}{4\times 3}=\frac{11}{3}$
Hence, range of the function is $[\frac{11}{3},\infty ).$


Alternate Solution:
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 $[\frac{11}{3},\infty )$