f(x) looks scary. It is not. It is a machine with a name. Click Next.
From the last lesson: the pizza shop. Now a candy shop opens too.
Pizza shop: y = 3x + 1 Candy shop: y = 5x + 2
Your friend asks: "What is y when x is 4?" … WHICH y?! Both shops use the same letter. Confusing!
The fix: give each machine a NAME.
p(x) = 3x + 1 c(x) = 5x + 2
p for pizza. c for candy. Now nobody is confused.
Three symbols, three jobs.
Feed it a number. p(2) means: put 2 everywhere the x was.
p(2) = 3 × 2 + 1 = 7
p(2) = 7 says everything in one tiny sentence: machine p, fed 2, gives out 7. The letter y could never tell you the input!
Feed the SAME number to both machines.
p(4) = 3×4+1 = 13 c(4) = 5×4+2 = 22
Same input, different machines, different answers. The name tells you which machine worked.
So what is f? Just the most popular machine name. When a book has only one machine, it lazily calls it f (for "function").
f(x) = 3x + 1 is the pizza rule with a boring name.
Books also use g and h. Same idea: name(input) = recipe.
f(x) is just y with a name tag.
y = 3x + 1 and f(x) = 3x + 1 are the SAME rule.
And f(2) = 7 is the point (2, 7) on the graph. Input right, output up.
Your turn: p(x) = 3x + 1. Find p(10) in your head… then click Next.
p(10) = 3 × 10 + 1 = 31 ✓ (the point (10, 31))
f(x) is NOT f times x. The parentheses here are a mouth, not multiplication. f(2) means "feed 2 to machine f" — never "f × 2".
Second trap: in f(2) = 7, students grab the wrong number. 2 is the input (right on the graph). 7 is the output (up on the graph). "What is f(2)?" wants the 7.
Name(food) = recipe. f(2) = 7: feed 2, get 7.
The candy machine is c(x) = 5x + 2. What is c(3)?
Answer: (A) 17. Feed 3 into the machine: c(3) = 5 × 3 + 2 = 15 + 2 = 17.
(B) 53 is the "glue the numbers" trap. (C) is the "f times x" trap. (D) 10 forgot to multiply first.
In a sentence: 3 candies cost $17 with the bag. The point (3, 17).