Question

Use the given graph of \(f\) to find a number \(\delta \) such that if \(0 < \left| {x - 3} \right| < d\), then \(\left| {f\left( x \right) - 2} \right| < 0.5\)

Short answer

The number is \(0.4\).

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 \(\left| {f\left( x \right) - 2} \right| < 0.5\) as:

\(\begin{aligned} - 0.5 < f\left( x \right) - 2 < 0.5\\ - 0.5 + 2 < f\left( x \right) < 0.5 + 2\\1.5 < f\left( x \right) < 2.5\end{aligned}\)

So, \(1.5 < f\left( x \right) < 2.5\).

Observe the graph

It is observed from the graph that the condition \(1.5 < f\left( x \right) < 2.5\) is satisfied when \(x\) lies between \(2.6\) and \(3.8\), that is, \(2.6 < x < 3.8\).

This implies that \(\delta = {\rm{min}}\left\{ {3 - 2.6,3.8 - 3} \right\}\).

Obtain the value of \(\delta \)

Solve the obtained condition.

\(\begin{aligned}\delta &= {\rm{min}}\left\{ {3 - 2.6,3.8 - 3} \right\}\\ &= {\rm{min}}\left\{ {0.4,0.8} \right\}\\ &= 0.4\end{aligned}\)

The number \(\delta \) that satisfy the given condition is \(0.4\).