Inverse Functions (Grade 12) — Worksheet & Memo

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Grade 12 · Functions

Inverse Functions (Grade 12)

EXAM-STYLE CLASS TEST

Marks

Duration

55 minutes

Questions

Name:

Class:

Date:

Mark

/ 35

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
The inverse of a function is found by …
(1)
A)swapping x and y, then making y the subject
B)changing the sign of y
C)reflecting in the y-axis
D)taking the reciprocal of the function
1.2
The graph of f⁻¹ is the reflection of f in the line …
(1)
A)y = x
B)y = −x
C)the x-axis
D)the y-axis
1.3
The inverse of f(x) = 3x − 6 is …
(1)
A)x + 63
B)x − 63
C)13x − 6
D)3x + 6
1.4
The inverse of f(x) = x² for x ≥ 0 is …
(1)
A)√x
B)−√x
C)x²
D)1x²
1.5
Why must the domain of f(x) = x² be restricted before it has an inverse function?
(1)
A)without the restriction the inverse fails the vertical line test
B)x² is undefined for negative x
C)x² has no y-intercept
D)the inverse would otherwise be a straight line
1.6
If f(x) = 2^x, then f⁻¹(x) = …
(1)
A)log₂x
B)2^−x
C)x²
D)12^x
1.7
The DOMAIN of f⁻¹ is …
(1)
A)the range of f
B)the domain of f
C)always all real x
D)always x > 0
1.8
If (3 ; 7) lies on f, then f⁻¹ passes through …
(1)
A)(7 ; 3)
B)(3 ; 7)
C)(−3 ; −7)
D)(−7 ; 3)
1.9
The inverse of y = ax is …
(1)
A)xa
B)ax
C)ax
D)−ax
1.10
A function's inverse is itself a function only when the original is …
(1)
A)one-to-one
B)many-to-one
C)always increasing
D)continuous

Answer grid — circle your answers for Question 1

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[12 MARKS]

The graph of f(x) = log₄ x is drawn. B(k ; 2) is a point on f.

2.1
Calculate the value of k, and hence write down the coordinates of the point at which f cuts the x-axis.
(3)
2.2
Determine the values of x for which −1 ≤ f(x) ≤ 2.
(3)
2.3
Write down the equation of f⁻¹, the inverse of f, in the form y = …, and state its domain.
(3)
2.4
For which values of x will x · f⁻¹(x) ≤ 0?
(3)

Question 3

[13 MARKS]

Given g(x) = 2x² for x ≥ 0.

3.1
Determine the equation of g⁻¹ in the form y = …
(4)
3.2
Write down the domain and the range of g⁻¹, and say where each one comes from.
(3)
3.3
Explain why the inverse of g would NOT be a function if the domain of g were not restricted.
(3)
3.4
Describe the geometric relationship between the graphs of g and g⁻¹, and hence determine the coordinates of the point, other than the origin, at which they intersect.
(3)

TOTAL: 35 marks

This question paper consists of 3 questions.