SME
Grade 11 Math · Algebra 2
Function operations, composition and inverses
Use this for “f(x) = x² − 3x and g(x) = 2x + 1. Find f(g(x)).”
Chart
f(1) = 0
This is an example. Type your own values to start a new problem.
f(x)
=
p
x²
+
(
q
)
x
+
r
g(x)
=
m
x
+
c
f(g(x))
=
A
x²
+
(
B
)
x
+
(
C
)
x
Input
example
y
f(g(x))
worked out (not typed)
Formulas
A
=
p
×
m
2
4
=
1
×
2
2
B
=
2
×
p
×
m
×
c
+
q
×
m
−2 = 2 × 1 × 2 × 1 + (−3) × 2
C
=
p
×
c
2
+
q
×
c
+
r
−2
=
1
×
1
2
+
(−3)
×
1
+
0
y
=
A
×
x
2
+
B
×
x
+
C
0
=
4
×
1
2
+
(−2)
×
1
+
(−2)
Assumptions
•
f(g(x)) puts g(x) wherever f has x: here f(x) = px² + qx + r and g(x) = mx + c.
•
Work inside out: g first, then f.
•
f(g(x)) and g(f(x)) are usually different.
Step-by-step
WE KNOW
p = 1, q = −3, r = 0, m = 2, c = 1, x = 1
FIND
x² coefficient of f(g(x)) (A), x coefficient of f(g(x)) (B), constant of f(g(x)) (C), f(g(x)) (y)
Step 1 · Find x² coefficient of f(g(x)): A
Use A = p × m²
p
(
mx
+
c
)
2
starts
with
p
×
m
2
x
2
.
A
=
1
×
2
2
A
=
1
×
4
A
=
4
Step 2 · Find x coefficient of f(g(x)): B
Use B = 2 × p × m × c + q × m
The
x
terms:
p
×
2mcx
from
p
(
mx
+
c
)
2
,
and
q
×
mx
from
q
(mx
+
c
).
B
=
2
×
1
×
2
×
1
−
3
×
2
B
=
4
−
6
B
=
−2
Step 3 · Find constant of f(g(x)): C
Use C = p × c² + q × c + r
The
constants:
f
(
c
),
since
g
(0)
=
c
.
C
=
1
×
1
2
−
3
×
1
+
0
C
=
1
−
3
+
0
f
(
g
(
x
))
=
f
(
2
x
+
1)
=
4
x
²
−
2
x
−
2
C
=
−2
Step 4 · Find f(g(x)): y
Use y = A × x² + B × x + C
Evaluate
the
combined
function
at
x
.
y
=
4
×
1
2
−
2
×
1
−
2
y
=
4
−
2
−
2
y
=
0
Check
Put the numbers back in. Both sides should match.
4
=
1
×
2
2
✓
−2 = 2 × 1 × 2 × 1 − 3 × 2 ✓
−2
=
1
×
1
2
−
3
×
1
+
0
✓
0
=
4
×
1
2
−
2
×
1
−
2
✓
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Add, subtract, multiply and divide functions
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