A composite function applies one function to the output of another — a chained model. The notation \( (f \circ g)(x) = f(g(x)) \) means “do \( g \) first, then \( f \)”.
TI-84 Plus CE
Store the inner function in
Y1, the outer in
Y2
On the home screen type
Y2(Y1(3)) for \( (f\circ g)(3) \)
(
[VARS] → Y-VARS to paste \( Y1, Y2 \))
Or just evaluate the inner value first, then the outer.
TI-Nspire CX II
Define
g(x):=... and
f(x):=... with
[ctrl][:=] or
Define
Then type
f(g(3)) on a Calculator page
For an algebraic form, type
f(g(x))
Define functions once, reuse for any input.
Casio fx-CG50
[MENU] → Graph: enter
Y1,
Y2
On Run-Matrix type
Y2(Y1(3)) (paste with
[VARS] → GRAPH)
Or evaluate the inner value, then substitute
Keep the inner function in the lower Y-slot.
Worked Example
Let \( f(x)=2x+1 \) and \( g(x)=x^2 \). Find \( (f \circ g)(3) \).
Inner first: \( g(3)=3^2=9 \)
Then outer: \( f(9)=2(9)+1=19 \)
Answer: \( (f \circ g)(3)=19 \)
GDC: with \( Y1=x^2 \), \( Y2=2x+1 \), type Y2(Y1(3)) → 19.
✗
Common error: doing \( g(f(x)) \) by mistake. The inner function (closest to \( x \)) is applied first: \( (f\circ g)(x)=f(g(x)) \), so \( g \) goes first, not \( f \).
A transformation changes a graph’s position or shape. Changes outside the function (\( y \)) act vertically and behave as expected; changes inside the function (\( x \)) act horizontally and are reversed.
TI-84 Plus CE
Enter the original in
Y1
Enter the transform referencing it, e.g.
Y2 = 3*Y1(X-1)-2
[GRAPH] to compare both curves
Use [2ND][TRACE] → value to check mapped points.
TI-Nspire CX II
On a Graphs page define
f1(x)=...
Then
f2(x)=3·f1(x-1)-2 to plot the transform
Drag or use
[menu] → Trace to read points
Reference \( f1 \) so both update together.
Casio fx-CG50
[MENU] → Graph: enter
Y1
Enter
Y2 = 3 Y1(X-1) - 2 (paste \( Y1 \) via
[VARS] → GRAPH)
[F6] DRAW to overlay both
Use [SHIFT][F1] Trace to inspect.
Worked Example
The point \( (2, 5) \) lies on \( y = f(x) \). Find its image on \( y = 3f(x - 1) - 2 \).
Horizontal: shift right 1 ⇒ \( x: 2 \to 2 + 1 = 3 \)
Vertical: stretch \( \times 3 \) then down 2 ⇒ \( y: 5 \to 3(5) - 2 = 13 \)
Answer: image point \( (3, 13) \)
GDC: plot \( Y1 \) and \( Y2=3Y1(X-1)-2 \), trace to confirm.
✗
Common error: reading horizontal shifts backwards. \( y = f(x - 1) \) moves the graph right by 1 (not left), and \( y = f(2x) \) compresses horizontally by factor \( \tfrac{1}{2} \) (not stretches). Inside the bracket, everything is reversed.
▶
Half-life & doubling (GDC-first): set the model equal to half (or double) the starting amount and solve by graph intersection — plot the model and the target line, then find the intersection \( x \).
TI-84 Plus CE
Fit data:
[STAT] → CALC →
ExpReg (gives \( y=a\cdot b^{x} \))
Solve: store model in
Y1, target in
Y2
[2ND][TRACE] →
5:intersect for the time
Or [MATH] Solver for \( Y1 = \text{value} \).
TI-Nspire CX II
Fit:
[menu] → Statistics → Stat Calculations →
Exponential Regression
Solve:
nSolve(k·a^x + c = value, x)
Or graph the model and target, use Intersection Point(s)
nSolve returns the time directly.
Casio fx-CG50
Fit:
[MENU] → Statistics →
[F2] CALC →
Exp regression
Solve: Graph the model and target line
[SHIFT][F5] G-Solv →
INTSECT
Or use Equation → Solver for \( f(x)=\text{value} \).
Worked Example
A sample has mass \( M = 200(0.5)^{t/8} \) grams, \( t \) in hours. Find the half-life.
Half-life: solve \( 200(0.5)^{t/8} = 100 \Rightarrow (0.5)^{t/8} = 0.5 \)
So \( \dfrac{t}{8} = 1 \Rightarrow t = 8 \). (Check \( M=50 \): \( (0.5)^{t/8}=0.25 \Rightarrow t=16 \).)
Answer: half-life \( = 8 \) hours
GDC: intersect \( Y1=200(0.5)^{X/8} \) with \( Y2=100 \) → \( X=8 \).
✗
Common error: ignoring the asymptote \( c \). For \( f(x)=k\,a^{x}+c \) the amount does NOT halve to a true half — halve the part above \( c \), i.e. solve \( f(x)=c+\tfrac{1}{2}(f(0)-c) \) when a vertical offset is present.
TI-84 Plus CE
Fit data:
[STAT] → CALC →
Logistic
(returns \( y = c/(1 + a e^{-bx}) \), so \( L=c \))
Evaluate / solve via
Y1 and
intersect
Read \( L \) (carrying capacity) off the fit.
TI-Nspire CX II
Fit:
[menu] → Statistics → Stat Calculations →
Logistic Regression (d=0)
Read \( L \) as the numerator coefficient
Solve a time with
nSolve(...) or graph intersection
Plot to see the S-curve flatten at \( L \).
Casio fx-CG50
Fit:
[MENU] → Statistics →
[F2] CALC →
Logistic
The fit gives \( L \), \( C \), \( k \) (form \( L/(1+Ce^{-kx}) \))
Solve via G-Solv
INTSECT
The horizontal asymptote is \( y = L \).
Worked Example
A population is modelled by \( P = \dfrac{10000}{1 + 49 e^{-0.5t}} \), \( t \) in years. State the carrying capacity and find \( P \) when \( t = 10 \).
Carrying capacity \( = L = 10000 \) (limit as \( t \to \infty \)).
\( P(10) = \dfrac{10000}{1 + 49 e^{-5}} = \dfrac{10000}{1 + 49(0.006738)} \approx \dfrac{10000}{1.3302} \approx 7518 \)
Answer: carrying capacity \( 10000 \); \( P(10) \approx 7518 \)
GDC: store the model in \( Y1 \), evaluate at \( X=10 \) → 7518.
TI-84 Plus CE
Fit data:
[STAT] → CALC →
SinReg (set RADIAN mode first)
Returns \( y = a\sin(bx + c) + d \)
Solve a time via
Y1 &
intersect
Amplitude \( =|a| \), period \( =2\pi/b \).
TI-Nspire CX II
Fit:
[menu] → Statistics → Stat Calculations →
Sinusoidal Regression
Set Angle = Radian in Document Settings first
Solve with
nSolve(...) or graph intersection
Read \( a, b, c, d \) from the fit output.
Casio fx-CG50
Fit:
[MENU] → Statistics →
[F2] CALC →
Sin regression
Set Angle = Rad in SET UP (
[SHIFT][MENU]) first
Solve via G-Solv
INTSECT
Period \( =2\pi/b \); axis \( y=d \).
Worked Example
Tide height \( h = a\sin\big(b(t - c)\big) + d \) (m) has max \( 8 \), min \( 2 \), period \( 12 \) h, first max at \( t = 3 \). Find \( a, b, c, d \).
\( a = \dfrac{8 - 2}{2} = 3 \); \( d = \dfrac{8 + 2}{2} = 5 \); \( b = \dfrac{2\pi}{12} = \dfrac{\pi}{6} \)
A sine peaks a quarter-period (\( 3 \) h) after \( t = c \). Max is at \( t = 3 \), so \( c = 3 - 3 = 0 \). Check: \( h(3) = 3\sin\!\big(\tfrac{\pi}{6}(3)\big) + 5 = 3(1) + 5 = 8 \) ✓
Answer: \( a = 3 \), \( b = \dfrac{\pi}{6} \), \( c = 0 \), \( d = 5 \); principal axis \( h = 5 \) m
GDC: plot \( Y1 = 3\sin(\tfrac{\pi}{6}X) + 5 \) in radian mode; G-Solv MAX confirms 8 m at \( X=3 \).
▶
Set RADIAN mode before any SinReg / sinusoidal work — in degree mode \( b \) and the period come out as \( \dfrac{360^\circ}{b} \) and answers will not match a radian model. Find amplitude/axis from max & min; find \( b \) from the period.