Use the given graph off to find a number δ such that if |x−1|&lt;δ then |f(x)−1|&lt;0.2

Question

Use the given graph off to find a number δ such that if |x−1|&amp;lt;δ then |f(x)−1|&amp;lt;0.2

Philip Williams · Accepted Answer

|X|&amp;lt;a&amp;nbsp;implies&amp;nbsp;−a&amp;lt;X&amp;lt;a&amp;nbsp;Simplify by adding 1 to each|f(x)−1|&amp;lt;0.2&amp;nbsp;−0.2&amp;lt;f(x)−1&amp;lt;0.2&amp;nbsp;0.8&amp;lt;f(x)&amp;lt;1.2&amp;nbsp;The graph shows that if&amp;nbsp;0.8&amp;lt;f(x)&amp;lt;1.2&amp;nbsp;x must verify&amp;nbsp;0.7&amp;lt;x&amp;lt;1.1&amp;nbsp;On the other hand, x is required to confirm this inequality:|x−1|&amp;lt;δ&amp;nbsp;−δ&amp;lt;x−1&amp;lt;δ&amp;nbsp;If we can find&amp;nbsp;δ&amp;nbsp;that verify&amp;nbsp;−0.3≤−δ&amp;lt;x−1&amp;lt;δ≤0.1, we can make sure that&amp;nbsp;0.7&amp;lt;x&amp;lt;1.1&amp;nbsp;so&amp;nbsp;δ&amp;nbsp;must verify two conditions:&amp;nbsp;δ≤0.1&amp;nbsp;and&amp;nbsp;−0.3≤−δ&amp;nbsp;which means that&amp;nbsp;δ≤0.1&amp;nbsp;and&amp;nbsp;0.3≥δ&amp;nbsp;0.7&amp;lt;x&amp;lt;1.1&amp;nbsp;0.7−1&amp;lt;x−1&amp;lt;1.1−1&amp;nbsp;−0.3&amp;lt;x−1&amp;lt;0.1&amp;nbsp;We can choose&amp;nbsp;δ=0.1, so to be sure two conditions are satisfied, but if you want, you can choose any value.

Toni Scott · Accepted Answer

You can find it using the graph
1. Find the intervals where $f (x)$ and $x$ must be. In our case for $f (x) = [0.8, 1.2]$ , for $x = [0.7, 1.1]$
2. Find the middle of the first interval. We have $1$ $(\frac{0.8 + 1.2}{2} = 1)$
3. Find $x_{0}$ , wheree $f (x_{0}) Z = d \leq= 1, w e h a v e x_{0} = 1$
4. Calculate the distance between $x_{0}$ and the endpoints of the second interval, in our case: $| x_{0} - 0.7 | = | 1 - 0.7 | = 0.3$ and $| x_{0} - 1.1 | = | 1 - 1.1 | = 0.1$
5. $δ$ is the minimum of these two values, thus it's $0.1$

Use the given graph off to find a number \delta

Answered question

Answer & Explanation

New Questions in Algebra I