SME
Grade 11 Math · Problem type
The inverse of a linear function
Use this for “Find f⁻¹(x) for f(x) = 3x − 6, and check f⁻¹(f(x)) = x.”
Chart
f(4) = 6
The inverse is the reflection across the line y = x
Slope of f⁻¹: 1/3
Intercept of f⁻¹: 2
Back to x, f⁻¹(y): 4
This is an example. Type your own values to start a new problem.
f(x)
=
a
x
+
(
b
)
f⁻¹(x)
=
1
3
m
x
+
n
x
Input of f
example
y
f(x), the input of f⁻¹
calculated
X
Back to x, f⁻¹(y)
worked out (not typed)
Formulas
y
=
a
×
x
+
b
6 = 3 × 4 + (−6)
m
=
1
a
1
3
=
1
3
n
=
−
b
a
2
=
−
−6
3
X
=
m
×
y
+
n
4
=
1
3
×
6
+
2
Assumptions
•
Swap x and y in y = ax + b, then solve for y: f⁻¹(x) = (x − b) ÷ a = mx + n.
•
(x, y) on f is (y, x) on f⁻¹: the two graphs are reflections over y = x.
•
a is not 0, or f is flat and has no inverse.
Step-by-step
WE KNOW
a = 3, b = −6, x = 4
FIND
f(x), the input of f⁻¹ (y), slope of f⁻¹ (m), intercept of f⁻¹ (n), back to x, f⁻¹(y) (X)
Step 1 · Find f(x), the input of f⁻¹ (y)
Use y = a × x + b
Put
x
into
f.
y
=
3
×
4
−
6
y
=
12
−
6
y
=
6
Step 2 · Find slope of f⁻¹ (m)
Use m = 1 ÷ a
Swap
x
and
y
in
y
=
ax
+
b
;
y
is
then
x
−
b
a
,
so
its
slope
is
1
a
.
m
=
1
3
m
=
1
3
Step 3 · Find intercept of f⁻¹ (n)
Use n = −b ÷ a
Solve
x
=
ay
+
b
for
y
:
take
away
b
,
then
divide
by
a
.
The
−
b
is
divided
by
a
too.
n
=
−
−6
3
x
=
3
y
−
6
x
+
6
=
3
y
y
=
x
+ 6
3
=
1
3
x
+
2
n
=
6
3
n
=
2
Step 4 · Find back to x, f⁻¹(y): X
Use X = m × y + n
Check:
put
y
=
f
(
x
)
into
f
-1
.
It
lands
back
on
x
,
so
f
-1
undoes
f.
X
=
1
3
×
6
+
2
X
=
2
+
2
X
=
4
Check
Put the numbers back in. Both sides should match.
6 = 3 × 4 − 6 ✓
1
3
=
1
3
✓
2
=
−
−6
3
✓
4
=
1
3
×
6
+
2
✓
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Inverse on a restricted domain
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Related lessons
Function operations, composition and inverses
Main lesson
Add, subtract, multiply and divide functions
Use this for “f(x) = 2x − 1 and g(x) = x + 4. Find (f + g)(3), (f − g)(3), (f · g)(3) and (f ÷ g)(3).”
Inverse on a restricted domain
Use this for “f(x) = 2(x − 1)² + 3 for x ≥ 1. Find f⁻¹(11) and f⁻¹(x).”