Mathematics
9. For the functions: √
f : R → R, f (x) = x2 + 2x, g : [0, 2] → R, g(x) = x
a. Find the rule for (f + g)(x)
b. Sketch the graph of (f + g)(x) on a set of axes.
c. State the range
d. Define the function of (f + e)(x)
10. Two functions p and q are continuous over their domains, which are [−2, 3] and
(−1, 3] respectively. The domain of the sum function p + q is:
A. [−2, 3]
B. [−2, −1) ∪ [0, 5]
C. [−2, −1] ∪ (−1, 3] ∪ [0, 5]
D. [−1, 3]
E. (−1, 3)
1E Composite Functions (1 Session)
Key Points
1. The composite function of g with f is written as g ◦ f , read as ’composite function
of f followed by g’.
2. g ◦ f (x) = g(f (x))
3. For g ◦ f (x) to be defined, the range of f must be a subset of the domain of g, i.e.,
ran f ⊆ dom g.
4. dom(g ◦ f ) = dom f .
5. In general, g ◦ f ̸= f ◦ g.
Anandakumar Arulanantham Anujan Page 6
Mathematics
1. For the functions g(x) = log2 (2x − 1), x > 12 , and f (x) = x + 1, x ∈ R,
a. State whether g ◦ f and f ◦ g are defined. Explain.
b. State the rule(s) of the composite function(s) which is/are defined.
c. State the domain(s) of the composite function(s).
√
2. For the functions f (x) = x2 − 2 and g(x) = x + 1,
a. State the implied domain and range for each of the functions above.
b. Define the composite function f (g(x)).
c. Explain why g(f (x)) is undefined.
d. Let f : D → R, f (x) = x2 − 2 where D is the maximal domain for g(f (x)) to
be defined. Find the maximal domain D.
Anandakumar Arulanantham Anujan Page 7
Mathematics
3. (2022 MM EXAM 2)
Consider the composite function g(x) = f (sin(2x)), where the function f (x) is an
unknown but differentiable function for all values of x.Use the following table of
values of f and f ′ :
√ √
π 2 3
x 2 2 2
f (x) 2 5 3
f ′ (x) 7 0 1
2
π
a. Find the value of g 3
.
b. Show that g ′ π
= 21 .
6
1F Inverse Functions (0.5 Session)
Key Points of Inverse Functions
• A function f (x) has an inverse function only if f (x) is a one-to-one function,
denoted as f −1 (x).
• The relationships between the domains and ranges of f (x) and f −1 (x) are shown
below:
f (x) f −1 (x)
domain range
range domain
• The graphs of f (x) and f −1 (x) are symmetrical about the line y = x.
• To find f −1 (x) given f (x), follow these steps:
(a) Replace f (x) with y (optional), then swap x with y.
(b) Make y the subject and replace y with f −1 (x).
• f (f −1 (x)) = f −1 (f (x)) = x.
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Mathematics
1
1. For the function f (x) = (x−2)2
,
a. Find its domain and range.
b. Explain why f −1 (x) does not exist for the domain found in Q1a.
c. Define a maximum suitable restriction for f to make f −1 (x) exist over (a, ∞).
Find the minimum value of a.
d. Hence, find the rule of f −1 (x).
e. Define the function f −1 (x) using function notation.
f. Sketch the graphs of f (x) and f −1 (x) for x ∈ (2, ∞) on the same set of
axes, labelling all key points with coordinates. You don’t need to label the
coordinates for the intersection point.
Anandakumar Arulanantham Anujan Page 9
Mathematics
2. Evaluate:
(i) (f ◦ f −1 )(4).
(ii) f −2 (f (4)).
(iii) f (f −1 (1)).
√
3. (2020 MM EXAM 1) Let f : [0, 2] → R, where f (x) = √1 x.
2
a. Find the domain and the rule for f −1 , the inverse function of f .
b. The graph of y = f (x), where x ∈ [0, 2], is shown on the axes below.
c. On the axes above, sketch the graph of f −1 over its domain. Label the end-
points and points of intersection with f , giving their coordinates.
Anandakumar Arulanantham Anujan Page 10
