Question

Use the given graph of \(f\left( x \right) = \sqrt x \) to find a number \(\delta \) such that if \(\left| {x - 4} \right| < \delta \), then \(\left| {\sqrt x - 2} \right| < 0.4\)

Short answer

The number is \(1.44\).

Step-by-step solution

Solve the given condition

Apply the absolute property, that is, if \(\left| x \right| < a\) then \( - a < x < a\).

Rewrite the given condition \(\left| {\sqrt x - 2} \right| < 0.4\) as:

\(\begin{aligned} - 0.4 < \sqrt x - 2 < 0.4\\ - 0.4 + 2 < \sqrt x < 0.4 + 2\\1.6 < \sqrt x < 2.4\end{aligned}\)

Thus, \(1.6 < \sqrt x < 2.4\).

Observe the graph

It is observed from the graph and the obtained condition in step 1 that the left part will satisfy the condition \(\sqrt x = 1.6\) and right part will satisfy the condition \(\sqrt x = 2.4\).

Solve the obtained conditions.

\(\begin{aligned}x &= {\left( {1.6} \right)^2}\\ &= 2.56\end{aligned}\)

\(\begin{aligned}x &= {\left( {2.4} \right)^2}\\ &= 5.76\end{aligned}\)

Obtain the value of \(\delta \)

For the left side, the condition is \(\left| {x - 4} \right| < \left| {2.56 - 4} \right|\) and for right side is \(\left| {x - 4} \right| < \left| {5.76 - 4} \right|\). Simplify both the conditions as shown:

\(\begin{aligned}\left| {x - 4} \right| < \left| {2.56 - 4} \right|\\& = \left| { - 1.44} \right|\\ &= 1.44\\\left| {x - 4} \right| < 1.44\end{aligned}\)

And,

\(\begin{aligned}\left| {x - 4} \right| < \left| {5.76 - 4} \right|\\ &= \left| {1.76} \right|\\ &= 1.76\\\left| {x - 4} \right| < 1.76\end{aligned}\)

This implies that \(\delta = {\rm{min}}\left\{ {1.44,1.76} \right\}\), that is, \(\delta = 1.44\).

The number \(\delta \) that satisfy the given condition is \(1.44\), that is, \(\left| {x - 4} \right| < 1.44\).