Question

Use the given graph of f to find a number δ such that
if
|x − 1| < δ then |f(x) − 1| < 0.2

Answer 1

thats the graph. How to do this question?

Answer 2

you can interpret
|f(x) − 1| < 0.2
to mean
"the maximum distance of f(x) from 1 is less than 0.2"

another way is to say
|f(x) − 1| < 0.2
means
0.8 < f(x) < 1.2

the graph shows exactly this.

Answer 3

the graph also shows that
|f(x) − 1| < 0.2
if
0.7 < x < 1.1

i.e. if 0.7 < x < 1.1 then we know |f(x) − 1| < 0.2

however, that interval is not symmetric.
we want to find δ
|x − 1| < δ
or
1-δ < x < 1 + δ

notice if we use the interval 0.9 < x < 1.1
we are still guaranteed that |f(x) − 1| < 0.2

and using 0.9 < x < 1.1, we have a symmetric interval
with δ = 0.1

so one answer to the question (we could use a smaller interval, so in theory there are infinite number of answers): δ = 0.1