To write the equation [tex]\(6c = 2p - 10\)[/tex] in function notation where [tex]\(c\)[/tex] is the independent variable, we first need to solve for [tex]\(p\)[/tex] in terms of [tex]\(c\)[/tex].
Starting with the given equation:
[tex]\[6c = 2p - 10\][/tex]
1. Add 10 to both sides to isolate the term involving [tex]\(p\)[/tex]:
[tex]\[6c + 10 = 2p\][/tex]
2. Divide both sides by 2 to solve for [tex]\(p\)[/tex]:
[tex]\[p = 3c + 5\][/tex]
Now, we express this equation in function notation. Typically, function notation is written as [tex]\(f(x)\)[/tex], where [tex]\(x\)[/tex] is the independent variable. Here, since we are writing [tex]\(p\)[/tex] in terms of [tex]\(c\)[/tex], our function notation will denote [tex]\(p\)[/tex] as a function of [tex]\(c\)[/tex].
Therefore, the equation [tex]\(p = 3c + 5\)[/tex] can be written in function notation as:
[tex]\[f(p) = 3c + 5\][/tex]
Reviewing the provided options:
- [tex]\(f(c) = \frac{1}{3} p + \frac{5}{3}\)[/tex]
- [tex]\(f(c) = 3c + 5\)[/tex]
- [tex]\(f(p) = \frac{1}{3} p + \frac{5}{3}\)[/tex]
- [tex]\(f(p) = 3c + 5\)[/tex]
The correct option that matches the function notation we derived is:
[tex]\[f(p) = 3c + 5\][/tex]
So the corresponding correct choice is:
[tex]\[f(p)=3 c + 5\][/tex] which is the fourth option.
