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Question
4. If
show that fof (x) = x for all x ≠ 2/3 What is the inverse of f
Solution :
Hence, the given function f is invertible and the inverse of f is itself.
Question
7. Consider f : R→ R given by f(x) = 4x + 3 Show that f is invertible. Find the inverse of f.
Solution :
Question
8. Consider
f
: R
+
→ [4, ∞) given by f(
x
) =
x
2
+ 4 Show that f is invertible with the inverse
f
−1
of f given by
f
−1
(y) = √y - 4 where R
+
is the set of all non-negative real numbers.
Solution :
Question
9. Consider
f
: R
+
→ [−5, ∞) given by
f
(
x
) = 9
x
2
+ 6
x
− 5. Show that
f
is invertible with
Solution :
Question
10. Let f: X → Y be an invertible function. Show that f has unique inverse.
(Hint: suppose g
1
and g
2
are two inverses of f. Then for all y ∈ Y,
fog
1
(y) = IY(y) = fog
2
(y). Use one-one ness of f).
Solution :
Question
11. Consider f: {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f
−1
and show that (f
−1)−1
= f.
Solution :
Question
12. Let f: X → Y be an invertible function. Show that the inverse of f−1 is f, i.e., (f
−1
)
−1
= f.
Solution :
Let f: X → Y be an invertible function.
Then, there exists a function g: Y → X such that gof = IXand fog = IY.
Here, f
−1
= g.
Now, gof = IX and fog = IY
⇒ f
−1
of = IX and fof
−1
= IY
Hence, f
−1
: Y → X is invertible and f is the inverse of f
−1
i.e., (f−1)
−1
= f.
Question
13.
(A) 1/x
3
(
B)
x
3
Question
14.
Solution :
