I keep tripping over function notation and it’s driving me a little bananas. In my head, f(x) keeps looking like “f times x,” even though I know that’s not right. I’m trying to build a better mental picture. Is f like a machine where you toss in an input and it spits out a number? If so, I think I’m mixing up “double the input” vs “double the output.”
Here’s where I get stuck. Using a pizza analogy: is f(2x) like ordering one pizza that’s twice as big, while 2f(x) is like ordering two regular pizzas? They feel similar but not the same. With a concrete function, say f(x) = x^2 + 3x, I worked out:
– f(2x) = (2x)^2 + 3(2x) = 4x^2 + 6x
– 2f(x) = 2(x^2 + 3x) = 2x^2 + 6x
These don’t match, so I’m guessing f(2x) ≠ 2f(x) in general. But is there a quick way to know when they would match (if ever) without expanding everything each time?
Second snag: when I see f(x+h) − f(x), my brain really wants to split it like f(x) + f(h) − f(x) = f(h). That feels too convenient, and I’m pretty sure it’s wrong, but I keep making that mistake. With the same f(x) = x^2 + 3x, I tried: f(x+h) = (x+h)^2 + 3(x+h) = x^2 + 2xh + h^2 + 3x + 3h, so f(x+h) − f(x) = 2xh + h^2 + 3h. That seems to work, but I’m not confident I’m thinking about it the right way. Why can’t I just split f over the + like that?
Basically: how should I read f(2x), 2f(x), and f(x+h) so I stop thinking “multiplication” and start thinking “plug in the whole thing”? Any simple rule-of-thumb or everyday analogy would be amazing. Any help appreciated!
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3 Responses
I used to read f(x) as “f times x” too, so you’re not alone; what helped me was treating f as a recipe: whatever sits inside the parentheses is the entire ingredient you substitute everywhere x appears. With that mindset, f(2x) means “run the recipe on 2x” (one pizza, twice as big), while 2f(x) means “run the recipe on x, then double the result” (two regular pizzas). They only match for all x when the function scales perfectly with inputs, i.e., f(cx) = c f(x) for every c; in one variable that essentially means f(x) = kx (a straight line through the origin). A quick check: monomials scale like f(2x) = 2^n f(x) for x^n, so only degree-1 terms behave linearly; for polynomials, f(2x) = 2f(x) holds identically only if the polynomial is ax. For f(x+h) − f(x), read it as “how much the output changes when I nudge the input by h”; you can’t split it as f(x) + f(h) because f generally doesn’t distribute over addition (that property only holds for truly linear maps). Your computation is spot on: substituting x → x + h gives f(x+h) − f(x) = 2xh + h^2 + 3h = h(2x + h + 3), and that factor of h is exactly why this expression underlies the derivative. Rule of thumb: keep parentheses sacred and do whole-substitution-replace every x in the formula by the entire inside expression (2x, x+h, whatever), simplify, and only then apply any outside multipliers like the 2 in 2f(x). For a gentle walk-through of this notation with more examples, see https://www.khanacademy.org/math/algebra/algebra-functions.
Think of f as a machine: f(2x) feeds in a doubled input (one pizza twice as big) while 2f(x) doubles the output (two regular pizzas); they match when f is linear, e.g., f(x)=3x gives f(2x)=6x=2f(x), but not for f(x)=x^2+3x. And f(x+h)−f(x) means “plug in the whole x+h, then subtract,” not f(x)+f(h)−f(x); for instance with f(x)=x^2, f(x+h)−f(x)=2xh+h^2-nice quick refresher: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86/functions/e/evaluating-functions
Think of f(·) as “do the f‑procedure to whatever’s inside the parentheses.” Not multiply. So f(2x) means “everywhere the recipe uses x, use 2x instead.” But 2f(x) means “do the recipe first, then double the final number.” Your pizza instinct is fine: one double‑size pizza vs two regulars. They only match when the recipe scales perfectly with the input – basically f(x) = a·x. Any curve (like x^2) or an offset (+b) breaks it. Quick gut check: replace x with 2x in your head – if every term just doubles, you’re good; if anything quadruples, stays the same, etc., then no. A sloppy shortcut some folks use is “try x=1 and x=2; if f(2)=2f(1), maybe it holds” – but that can lie to you, so don’t trust it beyond a sniff test.
For f(x+h) − f(x), read it as “how much the output changes when I nudge the input by h.” You can’t split f(x+h) into f(x)+f(h) because f doesn’t distribute over + like multiplication does – only very special add‑up‑nicely recipes do that. For a straight‑line example f(x)=ax+b, what actually happens is f(x+h)=f(x)+f(h)−b; the +b gets counted twice if you try to split. Your expansion 2xh + h^2 + 3h is exactly the “new stuff” created by shoving in +h. Handy mental trick: treat f(□) like a slot – whatever’s in the slot gets used everywhere. And if h is tiny and you’re just estimating, keep the terms with a single h (like 2xh+3h) and ignore h^2; it’s like adding a little extra pepper to the soup – you taste the first pinch, not the tiny second‑order sprinkle.