Let's start with the given equation:
[tex]\[ 6c = 2p - 10 \][/tex]
We need to rewrite this equation in function notation, where [tex]\( c \)[/tex] is the independent variable.
First, let's solve for [tex]\( p \)[/tex] in terms of [tex]\( c \)[/tex]:
1. Add 10 to both sides:
[tex]\[ 6c + 10 = 2p \][/tex]
2. Divide both sides by 2 to isolate [tex]\( p \)[/tex]:
[tex]\[ p = \frac{6c + 10}{2} \][/tex]
3. Simplify the expression:
[tex]\[ p = 3c + 5 \][/tex]
Now, we need to write the equation in function notation. Since we expressed [tex]\( p \)[/tex] as the dependent variable in terms of [tex]\( c \)[/tex], let's convert this into a function. Although the problem seems to provide function notation in terms of [tex]\( p \)[/tex], we consider the general form [tex]\( f(p) \)[/tex] in terms of [tex]\( p \)[/tex].
Given that we already simplified [tex]\( p = 3c + 5 \)[/tex], let's rewrite it in function form:
[tex]\[ p = 3c + 5 \][/tex]
To write it in the function form f(p), we transform it directly from the expression [tex]\( p = 3c + 5 \)[/tex]:
[tex]\[ f(p) = 3c + 5 \][/tex]
Thus, the function in terms of [tex]\( p \)[/tex] is:
[tex]\[ f(p) = \frac{1}{3}p + \frac{5}{3} \][/tex]
Among the given choices, the one that matches this result is:
[tex]\[ \boxed{f(p) = \frac{1}{3} p + \frac{5}{3}} \][/tex]
