Substitute (a + 2) for every x: f(a + 2) = 3(a + 2) - 1 = 3a + 6 - 1 = 3a + 5. The entire expression (a + 2) replaces x; you cannot just add 2 to the output.
Question 2 True / False
f(x + 2) is the same as f(x) + 2 for any function f.
TTrue
FFalse
Answer: False
f(x + 2) changes the INPUT — the 2 is added before the rule is applied. f(x) + 2 changes the OUTPUT — the rule runs first, then 2 is added. For f(x) = x², f(x + 2) = (x+2)² = x² + 4x + 4, while f(x) + 2 = x² + 2. These are different.
Question 3 Short Answer
A student writes '2f(x) means the same as f(2x).' Give a counterexample using f(x) = x².
Think about your answer, then reveal below.
Model answer: At x = 3: 2f(3) = 2(9) = 18, but f(2·3) = f(6) = 36. They are not equal.
2f(x) multiplies the OUTPUT by 2, while f(2x) multiplies the INPUT by 2 before applying the rule. These are different operations and produce different results for any non-linear function.