Answer to Question #125172 in Calculus for EMMANUEL

Question

Two functions f and g are defined on the set of real numbers, R, by
f : x = x/2-x
g : x = px + q, where p and q are constants.
(i) Find g^−1
(ii) Given that g^−1(4) = 2 and g ◦ f(4) = −8, find the values of p and q

Answer 1

i)g^-1=inverting g(x)

 g(x)=px+q

y-q=px

g^-1=$(x-q)/p$

ii) Taking g ◦ f(4) = −8 to get the first equation;

g(4/2-4)=-8

 g(-2)=-8

p(-2)+q=-8

-2p+q=-8…….(i)

Taking g^-1 to get the second equation;

Y=(x-q)/p where y=2;

2p=4-q

2p+q=4………(ii)

Taking the two simultaneous equations and solving by elimination method;

-2p+q=-8

 2p+q= 4

-4p=-12

Hence p=3 and q=-2