Question

Find the range of the quadratic expression for real $x$

• A. $3x^2+2x+11$
• B. ${\sin }^{2}x+\sin x+2$
• C. $4^x-2^x+1$
• D. $4^x+2^2+1$

Answer 1

Answer: (A)

$3x^2+2x+11$

1. $3x^2+2x+11$

Range of $3x^2+2x+11:[Solution:f(x)\ge \frac{32}{3},Interval-notation:[\frac{32}{3},\infty \left)\right]$

For a parabola $a{x}^2+bx+c$ the vertex's x equals $\frac{-b}{2a}$

$x=\frac{-2}{2\cdot 3}$

$\Rightarrow x=\frac{-1}{3}$

$\Rightarrow y=3{\left(\frac{-1}{3}\right)}^2+2\left(\frac{-1}{3}\right)+11$

$\Rightarrow y=\left(\frac{32}{3}\right)$

Minimum $(\frac{-1}{3},\frac{32}{3})$

2.

Range of ${\sin }^{2}x+\sin x+2:[Solution:\frac{7}{4}\le f(x)\le 4,Interval-notation:[\frac{7}{4},4]]$

$x=\frac{\pi }{2}+2\pi n\Rightarrow {\sin }^{2}x+\sin x+2\Rightarrow y=4$

Maximum$(\frac{\pi }{2}+2\pi n,4)$

$x=\frac{7\pi }{6}+2\pi n\Rightarrow {\sin }^{2}x+\sin x+2\Rightarrow y=\frac{7}{4}$

Minimum$(\frac{7\pi }{6}+2\pi n,\frac{7}{4})$

$x=\frac{3\pi }{2}+2\pi n\Rightarrow {\sin }^{2}x+\sin x+2\Rightarrow y=2$

Minimum$(\frac{3\pi }{2}+2\pi n,2)$

$x=\frac{11\pi }{6}+2\pi n\Rightarrow {\sin }^{2}x+\sin x+2\Rightarrow y=\frac{7}{4}$

Minimum$(\frac{11\pi }{6}+2\pi n,\frac{7}{4})$

3.

Range of $4^x-2^x+1:[Solution:f(x)\ge \frac{3}{4},Interval-notation:[\frac{3}{4},\infty \left)\right]$

Inverse of $4^x-2^x+1:\frac{\ln \left(\frac{1\pm \sqrt{4x-3}}{2}\right)}{\ln 2}$

Domain of $\frac{\ln \left(\frac{1+\sqrt{4x-3}}{2}\right)}{\ln 2}:x\ge \frac{3}{4}$

Domain of $\frac{\ln \left(\frac{1+\sqrt{4x-3}}{2}\right)}{\ln 2}:\frac{3}{4}\le x\le 1$

Combine the intervals: $f\left(x\right)\ge \frac{3}{4}$

4.

Range of $4^x+2^2+1=4^x+5:[Solution:f(x)>5,Interval-notation:(5,\infty \left)\right]$

The range of an exponential function of the form $c\cdot n^{ax+b}+k$ is $f\left(x\right)>k$

$k=5$

$f\left(x\right)>5$