SME
Grade 11 Math · Problem type
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).”
Chart
f(3) = 11
Kept: x ≥ 1; the rest of the curve is dashed, and only the kept part is reflected
The inverse is the reflection across the line y = x
This is an example. Type your own values to start a new problem.
y
=
a
(
x
−
h
)²
+
k
Formulas
y
=
a
×
(
x
−
h
)
2
+
k
11
=
2
×
(3 − 1)
2
+
3
Assumptions
•
A whole parabola fails the horizontal line test, so keep only the half from the vertex on: x ≥ h.
•
On that half f only rises (or only falls when a < 0), so each y comes from one x.
•
Undo the steps in reverse order: take k away, divide by a, take the positive root, add h.
Step-by-step
WE KNOW
a = 2, h = 1, k = 3, y = 11
FIND
input, x ≥ h (x)
Step 1 · Find input, x ≥ h (x)
Use y = a × (x − h)² + k
The
inverse
undoes
f
in
reverse:
take
k
away,
divide
by
a
,
take
the
positive
root
(
x
≥
h
),
then
add
h
.
x
=
h
+
√(
y
−
k
a
)
x
=
1
+
√(
11 − 3
2
)
x
=
1
+
√(
8
2
)
x
=
1
+
√
4
x
=
1
+
2
x
=
3
(in
general,
f
-1
(
x
)
=
1
+
√(
x
− 3
2
))
Check
Put the numbers back in. Both sides should match.
11
=
2
×
(3 − 1)
2
+
3
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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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).”
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.”