$f(x+2)$ vs. $f(x)+2$
SAT glossary · Math
Definition
Not the same function: inside acts on the INPUT before the machine runs (x^2+4x+1), outside acts on the OUTPUT after (x^2-1). This is the transformation rule in disguise — one shifts left 2, the other shifts up 2 — and Desmos settles it in 10 seconds by graphing both.
$f(x+2)$ vs. $f(x)+2$ in plain English
Think of a function like f as a machine: you feed it a number, it spits out a new one. f(x+2) means you add 2 to the input BEFORE feeding it into the machine — so if f(x)=x^2+4x-3, then f(x+2) swaps every x for (x+2), giving x^2+4x+1 after you expand it. But f(x)+2 means you run the machine first and add 2 to whatever comes out AFTER — so it's just x^2+4x-3+2 = x^2+4x-1. Same machine, totally different move, totally different answer. Graphically, changing the input shifts the whole curve sideways; changing the output shifts it up or down. If you're ever unsure, plug in an actual number (like x=1) into both versions and see that they give different results — or graph both in Desmos and watch them land in different places.
Also called: x+2
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