1 Consider the following functions fz21 and gw a State the d

1. Consider the following functions: f(z)-2-1 and gw- a) State the domain of f and g. (4 points each) b) Find (f o g)(u) and state its domain (4 points) c) Find (go f)(x) and state its domain (4 points) d) Does g(v)=r1(y)? why or why not? (2 points)

Solution

f(x) = x/2 -1

g(y) = 1 / 2y - 1

a) domain of f(x) = all values of x ( - infinity , + infinity)

b) 1 / (2y-1)

denominator cannot be zero

so, 2y-1 not equal to zero

y not equal to 1/2

domain is all values of y and y not equal to 1/2

b) ( fog)(y) = 1/(2y-1)/2 -1

(fog)(y) = 1 / 2(2y-1) -1 = {1 / ( 4y - 2)} - 1

domain is all values of y and y not equal to 1/2

c) (gof)(x) = 1/ (2(x/2)-1 ) -1

(gof)(x) = 1/ x-2 -1 = 1 / x - 3

(gof)(x) = 1 / (x-3)

domain is all values of x and x not equal to 3

d) f(x) = x/2 - 1

y = x/2 -1

switch x and y to find teh inverse function

x = y/2 -1

2(x + 1) = y

f^-1(x) = 2(x+1)

f^-1 (y) = 2(y+1)

therefore , f^-1(y) is not equal to g(y)