Inverse Functions (Grade 12) — Worksheet & Memo

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Grade 12 · Functions and Graphs

Inverse Functions (Grade 12)

MEMORANDUM

Name:

Class:

Date:

Mark

/ 20

Answer all questions. Show all your working — marks are awarded for method as well as the final answer. Teacher copy: accept any correct equivalent method.

1
Determine the equation of h⁻¹, the inverse of h(x) = 3x − 6, in the form y = …
(2)
x = 3y − 6 → y = 1/3x + 2✓✓ (2)
2
Determine the inverse of f(x) = 2x².
(2)
x = 2y² → y = ±√(x/2)✓✓ (2)
3
Is the inverse of f(x) = 2x² a function? Give a reason.
(2)
No — because of the ±, one x-value gives two y-values, so it fails the vertical line test.✓✓ (2)
4
Restrict the domain of f(x) = 2x² so that f⁻¹ is a function.
(1)
x ≥ 0 (or x ≤ 0)✓ (1)
5
Determine the equation of k⁻¹ if k(x) = 2^x.
(2)
y = log₂ x✓✓ (2)
6
Write down the domain and range of k⁻¹ if k(x) = 2^x.
(2)
Domain x > 0; range y ∈ ℝ✓✓ (2)
7
The graph of f⁻¹ is the reflection of f in which line?
(1)
y = x✓ (1)
8
The point (2 ; 7) lies on f. Write down a point on f⁻¹.
(1)
(7 ; 2)✓ (1)
9
State the definition of a function.
(2)
A rule in which every input value has exactly ONE output value ✓✓(2)
10
Explain, using the definition, why the inverse of f(x) = 3x² is not a function unless the domain is restricted.
(3)
f is many-to-one — f(−2) and f(2) both give 12 ✓; reflecting swaps inputs and outputs, so one input would map to two outputs ✓; restricting to x ≥ 0 makes f one-to-one, so the inverse is a function ✓(3)
11
Which line test tells you whether the INVERSE of a graph will be a function?
(2)
The horizontal line test applied to the ORIGINAL graph ✓; if a horizontal line cuts it more than once, the inverse is not a function ✓(2)

TOTAL: 20 marks