Answer
Q:1
(a): Yes
(b): No
(c): Yes
(d): No
Work Step by Step
According to 0.4.1 Definition:
"If the functions f and g satisfy the two conditions
g(f(x)) = x for every x in the domain of f
f(g(y)) = y for every y in the domain of g
then we say that f is an inverse of g and g is an inverse of f or that f and g are inverse functions.
(a): Yes
Explanation:
Given that f(x)=4x and g(x)=1/4(x) = 4/x
f(g(x)) = 4(x/4) =x,
g(f(x)) = (4x)/4 =x
So, f and g are inverse functions.
(b): No
Explanation:
Given that f(x)=3x+1 and g(x)= 3x-1
f(g(x)) = 3(3x-1)+1 = 9x ,which is not equal to x
So, f and g are not inverse functions.
(c): Yes
Explanation:
Given that f(x)=\sqrt[3] (x-2) and g(x)= x^{3}+ 2
f(g(x)) = \sqrt[3] ((x^{3}+2)-2) =x,
g(f(x)) = \sqrt[3] (x-2)^{3} =x
So, f and g are inverse functions.
(d): No
Explanation:
Given that f(x)= x^{4} and g(x)= \sqrt[4] x
f(g(x)) = \sqrt[4] x^{4} ,which is not equal to x
So, f and g are not inverse functions.
