Solution 18829: Evaluating Composite Functions for a Specific Value of X on the TI-83 Family, TI-84 Plus Family, and TI-Nspire™ Handheld in TI-84 Plus Mode Graphing Calculators.
How can I evaluate the composition of two functions given a specific x-value on a TI-83 family, TI-84 Plus family, or TI-Nspire Handheld in TI-84 Plus Mode graphing calculator?
In order to evaluate the composition of two functions, given a specific value of x, refer to the following example.
Example: Find the composition of the following two functions when x = .5
f(x) = x2 + 3x - 2
g(x) = 2x + 3
Please Note: Many textbooks indicate function compositions using the notation f(g(x)) or (f ° g)(x).
Please Note: x) and Y2 = g(x)
1) Press [Y=] to go to the Y= Editor screen
2) Input x2 + 3x - 2 into the Y1 slot, then press [ENTER]
3) Input 2x + 3 into the Y2 slot, then press [ENTER]
4) Press [VARS] [Þ]  to select 1:Function from the Y-VARS menu
5) Press  to select 1:Y1 then press [(]
6) Press [VARS] [Þ]  to select 1:Function from the Y-VARS menu again
7) Press  to select 2:Y2 then press [)] [ENTER]
Please Note: The Y3 slot should read Y3 = Y1(Y2)
8) Press [Ü] [Ý] [Ý] [Ý] to select the '=' to the right of Y1, then press [ENTER] to disable the function from being graphed.
Please Note: The black square surrounding the '=' disappears once the function is disabled.
9) Press [ß] [ENTER] to disable the Y2 function from being graphed as well.
10) Press [GRAPH] to graph the composite function.
11) Press [TRACE] then input .5 and press [ENTER].
The coordinates are displayed, even if CoordOff format is selected. The y-value of 26 is the solution to the problem.
For additional instructions on evaluating functions and performing other calculations, refer to the online guidebook for your calculator model.
Please Note: TI-Nspire users may refer to the TI-84 Plus family guidebook.