R mutate multiple columns with if statement

I have data like this:

cols <- c("X01_01","X01_01_p", "X01_02","X01_02_p", "X01_03","X01_03_p", "X01_04", "X01_05","X01_06")
set.seed(111)
values <- replicate(9, sample(1:5, 4, replace = TRUE)) 
df <- as.data.frame(values)  

So my df looks like this:

    X01_01 X01_01_p X01_02 X01_02_p X01_03 X01_03_p X01_04 X01_05 X01_06
1      3        2      3        1      1        3      5      4      3
2      4        3      1        1      5        2      2      3      3
3      2        1      3        1      2        2      4      1      2
4      3        3      3        3      4        2      2      3      4

I have some columns to use for mutation (not all) and the names of the new columns.

cols_to_mutate <- c("X01_01_p","X01_02_p", "X01_03_p", "X01_04", "X01_05","X01_06")
new_cols <- c("X01_01_n","X01_02_n", "X01_03_n", "X01_04_n", "X01_05_n","X01_06_n")

Each mutation is the same:

• If the value is 1 or 2, the new value has to be 0
• If the value is 3, the new value has to be 0.5
• If the value is 4 or 5, the new value has to be 1

Ultimately my df looks like this:

    X01_01 X01_01_p X01_02 X01_02_p X01_03 X01_03_p X01_04 X01_05 X01_06 X01_01_n X01_02_n X01_03_n X01_04_n X01_05_n X01_06_n
1      3        2      3        1      1        3      5      4      3      0.0      0.0      0.5        1      1.0      0.5
2      4        3      1        1      5        2      2      3      3      0.5      0.0      0.0        0      0.5      0.5
3      2        1      3        1      2        2      4      1      2      0.0      0.0      0.0        1      0.0      0.0
4      3        3      3        3      4        2      2      3      4      0.5      0.5      0.0        0      0.5      1.0

In 'hard coding' I could write lots of lines like this:

df <- mutate(df, X01_01_n = ifelse(X01_01_p <= 2, 0, (ifelse(X01_01_p == 3, 0.5, 1))))
df <- mutate(df, X01_02_n = ifelse(X01_02_p <= 2, 0, (ifelse(X01_02_p == 3, 0.5, 1))))

But of course I am searching for a more fancy and quicker way to do this, but I searched and searched, but dit not find the solution. I tried:

df <- cbind(df,apply(df[,cols_to_mutate],2, function(x) if (x < 3) { 0} else if (x > 3) {1} else {.5}))

But this does not work. Any ideas would be great!!