Type the equation $f\left(x\right)\=\dots$

Context Panel: Assign Function

Figure 1.2.7(a) is an animation in which $f\left(x\right)\=\sqrt{x}$ is graphed in black, and $y\=\sqrt{3}$ is graphed in blue.

The slider in the animation toolbar controls the value of $\mathrm{\&varepsilon;}$. As the slider is moved past the first frame, red and green horizontal lines delineate an $\mathrm{\&varepsilon;}$-band around $y\&equals;\sqrt{3}$ and red and green vertical lines delineate the band $3-{\mathrm{\&delta;}}_L<x<3\&plus;{\mathrm{\&delta;}}_R$.

The red and green horizontal lines are drawn at $y\&equals;\sqrt{3}\pm \mathrm{\&varepsilon;}$, respectively, and the red and green vertical lines are drawn at the corresponding $x$-coordinates $x\&equals;f^{-1}\left(\sqrt{3}-\mathrm{\&varepsilon;}\right)$ and $x\&equals;f^{-1}\left(\sqrt{3}\&plus;\mathrm{\&varepsilon;}\right)$.

Write the equation $f\left(a-{\mathrm{\&delta;}}_L\right)\&equals;L-\mathrm{\&varepsilon;}$ 
Press the Enter key.

Context Panel: Solve≻Isolate Expression for≻${\mathrm{\&delta;}}_L$ 

Context Panel: Simplify≻Simplify

Write the equation $f\left(a\&plus;{\mathrm{\&delta;}}_R\right)\&equals;L\&plus;\mathrm{\&varepsilon;}$ 
Press the Enter key.

Context Panel: Solve≻Isolate Expression for≻${\mathrm{\&delta;}}_R$ 

Context Panel: Simplify≻Simplify

Clearly, ${\mathrm{\&delta;}}_L<{\mathrm{\&delta;}}_R$, but the choice $\mathrm{\&delta;}\left(\mathrm{\&varepsilon;}\right)\&equals;\mathrm{\&varepsilon;}$ is simpler.

Figure 1.2.7(b) suggests that ${\mathrm{\&delta;}}_L\&equals;2\sqrt{3}\mathrm{\&varepsilon;}-{\mathrm{\&varepsilon;}}^2\&gt;\mathrm{\&varepsilon;}$ for $0<\mathrm{\&varepsilon;}<2\sqrt{3}-1\doteq 2.46$.

To establish this inequality analytically, compare the left- and right-sides via the ratio