Math Snap

STEP 1

Assumptions1. The function $f$ has a known point $(1,1)$.
. The function $f$ intersects the line $y=0.8$ at $x=1.1$.
3. The function $f$ intersects the line $y=1.$ at $x=1.$.
4. We need to find a number $\delta$ such that if $|x-1|<\delta$ then $|f(x)-1|<0.$.

STEP 2

We are given that $f(x)$ intersects the lines $y=0.8$ and $y=1.2$ at $x=1.1$ and $x=1.2$ respectively. This means that $f(1.1) =0.8$ and $f(1.2) =1.2$.

STEP 3

We are looking for a $\delta$ such that if $|x-1|<\delta$ then $|f(x)-1|<0.2$. This means that the values of $f(x)$ should lie between $0.8$ and $1.2$ when $x$ is in the interval $(1-\delta,1+\delta)$.

STEP 4

Since $f(1.1) =0.8$ and $f(1.2) =1.2$, we can see that the values of $f(x)$ lie between $0.8$ and $1.2$ when $x$ is in the interval $(1.1,1.2)$.

STEP 5

Therefore, we can choose $\delta$ such that $(1-\delta,1+\delta)$ is a subset of the interval $(1.1,1.2)$.

The distance between $1$ and $1.1$ is $0.1$ and the distance between $1$ and $1.2$ is $0.2$. Therefore, we choose $\delta$ to be the smaller of these two distances, which is $0.1$.
So, $\delta =0.1$.
With this choice of $\delta$, if $|x-1|<\delta$ then $|f(x)-1|<0.2$.