killjoy1990xb9

2022-01-07

Use the given graph off to find a number $\delta$ such that if $|x-1|<\delta$ then $|f\left(x\right)-1|<0.2$

Philip Williams

$|X| implies $-a
Simplify by adding 1 to each
$|f\left(x\right)-1|<0.2$
$-0.2
$0.8
The graph shows that if $0.8 x must verify $0.7
On the other hand, x is required to confirm this inequality:
$|x-1|<\delta$
$-\delta
If we can find $\delta$ that verify $-0.3\le -\delta , we can make sure that $0.7
so $\delta$ must verify two conditions: $\delta \le 0.1$ and $-0.3\le -\delta$
which means that $\delta \le 0.1$ and $0.3\ge \delta$
$0.7
$0.7-1
$-0.3
We can choose $\delta =0.1$, so to be sure two conditions are satisfied, but if you want, you can choose any value.

Toni Scott

You can find it using the graph
1. Find the intervals where $f\left(x\right)$ and $x$ must be. In our case for $f\left(x\right)=\left[0.8,1.2\right]$, for $x=\left[0.7,1.1\right]$
2. Find the middle of the first interval. We have $1$ $\left(\frac{0.8+1.2}{2}=1\right)$
3. Find ${x}_{0}$, wheree
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.$\delta$ is the minimum of these two values, thus it's $0.1$

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