Composing functions on the home screen with Y-Vars

Function composition — (f ∘ g)(x) = f(g(x)) — is a standard algebra/precalc topic that students solve by hand. The calculator can do it directly once you know Y-Vars exists.

Setup

1. y= → type X^2 + 1 next to Y1=.
2. Arrow down to Y2=, type 2X - 3.

Now Y1 represents f(x) = x² + 1 and Y2 represents g(x) = 2x − 3.

Evaluating on the home screen

Quit back to the home screen (2nd + mode).

Type Y1(5) — but there's no Y1 key. You access it through the VARS menu:

vars → arrow right to Y-VARS → 1:Function… → 1:Y1. The token Y1 appears where your cursor was.

Continue: ( 5 ) enter. Result: 26 — because 5² + 1 = 26. ✓

Composing

To compute f(g(3)):

1. Type Y1( via vars → Y-VARS → 1 → 1.
2. Inside that, type Y2( the same way.
3. Type 3).
4. Close the outer paren: ).

Final expression: Y1(Y2(3)). Press enter. Calculator evaluates inside-out: Y2(3) = 2·3 − 3 = 3, then Y1(3) = 3² + 1 = 10. Returns 10.

Chaining deeper

There's no practical limit. Y1(Y2(Y3(Y2(5)))) works fine, subject to the home-screen character limit.

Symbolic composition (kind of)

The TI-84 isn't a CAS — it won't give you the composed formula symbolically. But you can verify a hand-derived composition by evaluating both the long form and the calculator form at several test points. If they match at 3 or 4 distinct inputs, the composition is almost certainly correct.

Plotting a composition

If you want the graph of f ∘ g, store it in a free slot: Y3 = Y1(Y2(X)). Then press graph. The calculator will plot the composed function over the current window.

Gotcha

The implicit multiplication rule trips people up. On the TI-84, Y1(X) means "evaluate Y1 at X" — it's a function call, not multiplication. If you actually want to multiply Y1 * X, write the operator explicitly.