How to normalise a dataset unevenly

So I have a dataset of life expectancy values between 30 and 100. I want to normalise them between 0-1, but I want to do it unevenly?

Basically I have four defined interval/breakpoint values creating 5 classes:

I can easily reclassify them into those five classes, but I don't know to calculate the normalised values using those breakpoints. All five classes have a range of 0.2, but the age range in each class will be different, e.g. class one, 0-0.2, has an age range of 27 years, but category two, 0.2-0.4, has a range of just 5 years.

Example data:

ages <- floor(runif(50, min = 30, max = 100))

Edit:

So on this graph, x is life expectancy in years, and y is the normalised values - I want to calculate the exact value of y for each x value.

One standard approach is to use logistic regression:

glmfit <- glm(Normalised_value~Age,data=df,family = binomial())
plot(glmfit)

predict(glmfit,newdata=data.frame(Age=30:100),type="response")

          1           2           3           4           5           6           7 
0.001038676 0.001270899 0.001554960 0.001902391 0.002327270 0.002846769 0.003481828 
          8           9          10          11          12          13          14 
0.004257952 0.005206175 0.006364213 0.007777825 0.009502425 0.011604952 0.014166036 
         15          16          17          18          19          20          21 
0.017282440 0.021069773 0.025665399 0.031231419 0.037957508 0.046063272 0.055799610 
         22          23          24          25          26          27          28 
0.067448397 0.081319557 0.097744402 0.117063988 0.139611296 0.165686430 0.195524878 
         29          30          31          32          33          34          35 
0.229260322 0.266885403 0.308216030 0.352866476 0.400242921 0.449561353 0.499891678 
         36          37          38          39          40          41          42 
0.550224198 0.599549040 0.646935614 0.691599168 0.732945010 0.770586518 0.804338777 
         43          44          45          46          47          48          49 
0.834193745 0.860284578 0.882846413 0.902179147 0.918615680 0.932497075 0.944154716 
         50          51          52          53          54          55          56 
0.953898634 0.962010835 0.968742352 0.974312922 0.978912346 0.982702836 0.985821857 
         57          58          59          60          61          62          63 
0.988385104 0.990489415 0.992215484 0.993630305 0.994789335 0.995738372 0.996515163 
         64          65          66          67          68          69          70 
0.997150770 0.997670717 0.998095963 0.998443694 0.998728001 0.998960424 0.999150415 
         71 
0.999305707 

plot(df$Age,df$Normalised_value)
lines(30:100,predict(glmfit,newdata=data.frame(Age=30:100),type="response"))

Original:

We can utilize splines::bs and lm, which creates piecewise linear spline regression:

library(splines)

spline_fit <- lm(Normalised_value ~ bs(Age, knots=c(57,62,66,71),degree=1),data=df)
newdata <- data.frame(Age=30:100)
newdata$normalized <- predict(spline_fit,newdata=newdata)
newdata

   Age   normalized
1   30 1.416396e-16
2   31 7.407407e-03
3   32 1.481481e-02
4   33 2.222222e-02
5   34 2.962963e-02
6   35 3.703704e-02
7   36 4.444444e-02
8   37 5.185185e-02
9   38 5.925926e-02
10  39 6.666667e-02
11  40 7.407407e-02
12  41 8.148148e-02
13  42 8.888889e-02
14  43 9.629630e-02
15  44 1.037037e-01
16  45 1.111111e-01
17  46 1.185185e-01
18  47 1.259259e-01
19  48 1.333333e-01
20  49 1.407407e-01
21  50 1.481481e-01
22  51 1.555556e-01
23  52 1.629630e-01
24  53 1.703704e-01
25  54 1.777778e-01
26  55 1.851852e-01
27  56 1.925926e-01
28  57 2.000000e-01
29  58 2.400000e-01
30  59 2.800000e-01
31  60 3.200000e-01
32  61 3.600000e-01
33  62 4.000000e-01
34  63 4.500000e-01
35  64 5.000000e-01
36  65 5.500000e-01
37  66 6.000000e-01
38  67 6.400000e-01
39  68 6.800000e-01
40  69 7.200000e-01
41  70 7.600000e-01
42  71 8.000000e-01
43  72 8.068966e-01
44  73 8.137931e-01
45  74 8.206897e-01
46  75 8.275862e-01
47  76 8.344828e-01
48  77 8.413793e-01
49  78 8.482759e-01
50  79 8.551724e-01
51  80 8.620690e-01
52  81 8.689655e-01
53  82 8.758621e-01
54  83 8.827586e-01
55  84 8.896552e-01
56  85 8.965517e-01
57  86 9.034483e-01
58  87 9.103448e-01
59  88 9.172414e-01
60  89 9.241379e-01
61  90 9.310345e-01
62  91 9.379310e-01
63  92 9.448276e-01
64  93 9.517241e-01
65  94 9.586207e-01
66  95 9.655172e-01
67  96 9.724138e-01
68  97 9.793103e-01
69  98 9.862069e-01
70  99 9.931034e-01
71 100 1.000000e+00