MULUNGUSHI UNIVERSITY
SCHOOL OF NATURAL AND APPLIED SCIENCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
MSM 111 - Mathematical Methods I
Tutorial Sheet 2 - 2022/2023 - Introduction to Functions
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√
1. Let g(x) = x2 − 4 and h(x) = x. Find
(a) (g ◦ h)(x) and its domain and range
(b) (h ◦ g)(x) and its domain and range
2. For each of the following functions, (i) list its domain and range and (ii) form the inverse functions f −1 and list the
domain and range of f −1 .
(a) f = {(1, 5), (2, 9), (5, 21)}
(b) f = {(0, 0), (2, 8), (−1, −1), (−2, −8)}
3. Verify that the two given functions are inverse of each other:
(a) f (x) = 5x − 9, g(x) =
(a) f (x) =
x−3
2 ,
x+9
5
(b) f (x) = x3 + 1, g(x) =
g(x) = 2x + 3
(b) f (x) =
1
x−1 ,
√
3
x−1
for x > 1, g(x) =
x+1
x
for x > 0.
4. For each given function, find f −1 and verify that (f ◦ f −1 )(x) = x and (f −1 ◦ f )(x) = x :
(a) f (x) =
2
x−1
for x > 1
(b) f (x) =
1−x
x ,
for x 6= 0
f (x) = 4x − 8.
5. If f (x) = 2x + 3 and g(x) = 3x − 5, find
(a) (f ◦ g)−1 (x)
(b) (f −1 ◦ g −1 )(x)
(c) (g −1 ◦ g −1 )(x)
6. Determine which of the following functions are even, odd or neither even nor odd:
(a) f (x) = 2x4 −4x+5
(b) f (x) = x2 −3x+5
(c) f (x) = 5x3 −4x
(d) f (x) =
√
(e) f (x) = − 9 − x2
3x−1
1−x2
7. Sketch the
√ graphs of each of the following functions involving radicals, on the same coordinate system as that of
f (x) = x and find the domain and range:
√
√
√
(a) f (x) = 3 − x
(b) f (x) = 2 − −x − 2
(c) f (x) = x2
q
√
√
(d) f (x) = −3 + x + 2
(e) f (x) = 2x − 1 + 3
(f) f (x) = −1 − 3 x − 21 .
8. Sketch the graphs of the following functions on the same axis as that of f (x) =| x |, and find the domain and range:
(a) f (x) =| x + 2 |
(b) f (x) =| 2x + 3 |
(d) f (x) =| 3x + 1 | − | 2x − 3 |
(g) f (x) =| 3x |
(c) f (x) =
1
3
| 4 − 3x | +3
(f) f (x) =| 2x − 1 | − | 3x + 1 |.
9. Solve each of the following
(a) | −2x − 1 |= 6
(b) | 2x + 1 |=| 4x − 3 |
(c)
10. Solve the following equations for x ∈ R.
√
√
√
(a) 2x − 3 = 1
(b) 3x − 8 − x + 2 = 0
−2
k+3
=5
(c)
√
(d)
x+1
x−1
=3
5x − 1 = −4
(e) | 2x − 1 | − | 3x + 1 |= 0
(d) | 2 − x |=
√
4−x
11. Find the set of values of x for which.
(a) 2x − 1 < 4(x − 3)
(e) | x − 2 |> −4
(i)
√
1−x>1
(b)
3
x−1
>1
(c)
(f) | 2x + 1 |>| x − 1 |
(j)
√
2x − 1 ≤
√
x−1
x+1
>2
(g) 2x+ | x |< 6
√
(k) − x + 1 < 3
2x − 1
1
(d)
3
x−2
(h)
(l)
<
x−1
x+2
√
√x−1
x+1
2
x+1
<2
>2