Question

Use the given graph of \(f\) to find a number \(\delta \) such that if \(\left| {x - 1} \right| < \delta \), then \(\left| {f\left( x \right) - 1} \right| < 0.2\).

Short answer

The number is \(0.1\).

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 \(\left| {f\left( x \right) - 1} \right| < 0.2\) as:

\(\begin{aligned} - 0.2 < f\left( x \right) - 1 < 0.2\\ - 0.2 + 1 < f\left( x \right) < 0.2 + 1\\0.8 < f\left( x \right) < 1.2\end{aligned}\)

So, \(0.8 < f\left( x \right) < 1.2\).

Observe the graph

It is observed from the graph that, the condition \(0.8 < f\left( x \right) < 1.2\) is satisfied when \(x\) lies between \(0.7\) and \(1.1\), that is, \(0.7 < x < 1.1\).

This implies that \(\delta = {\rm{min}}\left\{ {1 - 0.7,1.1 - 1} \right\}\).

Obtain the value of \(\delta \)

Solve the obtained condition as shown below:

\(\begin{aligned}\delta &= {\rm{min}}\left\{ {1 - 0.7,1.1 - 1} \right\}\\ &= {\rm{min}}\left\{ {0.3,0.1} \right\}\\ &= 0.1\end{aligned}\)

The number \(\delta \) that satisfy the given condition is \(0.1\).