Question

The range of the quadratic function $f\left(x\right)=x^2+x+1$ is given by

• A. $[\frac{1}{4},\infty )$
• B. $(-\infty ,\frac{1}{4}]$
• C. $[\frac{3}{4},\infty )$
• D. $(-\infty ,\frac{3}{4}]$

Answer 1

The correct option is C $[\frac{3}{4},\infty )$

For any quadratic expression $g\left(x\right)=a{x}^2+bx+c$, the graph is an upward opening parabola if $a>0$ and a downward opening parabola if $a<0$. Also its extreme point or vertex$\left(V\right)$ is given by $V:(\frac{-b}{2a},\frac{4ac-b^2}{4a})$

Comparing $f\left(x\right)=x^2+x+1$ with the above expression, we get $a=1>0,$
Hence the graph would be an upward opening parabola with vertex $\left(V\right)$ given by
$V\equiv (\frac{-\left(1\right)}{2\left(1\right)},\frac{4\left(1\right)\left(1\right)-(1{)}^2}{4\left(1\right)})\equiv (-\frac{1}{2},\frac{3}{4})$

Now since, the graph of the expression is an upward opening parabola with lowest point $\left(V\right)$ as $(-\frac{1}{2},\frac{3}{4})$. We can say that the range of $f\left(x\right)=x^2+x+1$ is $[\frac{3}{4},\infty )$