Determine the inverse in each of the following and prove the

Determine the inverse in each of the following and prove the result: f(x) = 4x + 3; g(x) = 5x/x - 2 h(x) = 2 Squareroot x + 1 + 3 Show that f(x) = 9/5 x + 32 and g(x) = 5/9 (x - 32) are inverse of each other. If f(x) = 2/x + 3 and g(x0 = 3/x - 4, find the domain of (f ocy g)(x). Let f(x) {-3, x 2 Find f(1), f(-3) and f(4) Sketch the GR and determine domain and Range of f(x)

Solution

To find inverse pplug y = f(x).Now plug x= y and y =x .Thensolve for y

y = f^-1(x)

5) a) f(x) = 4x +3

y = 4x +3

x = 4y +3

y = (x-3)/4

f^-1(x) = ( x-3)/4

To prove that it is the inerse : plug x= f(x) in f^-1(x) we get x >hence proved

f^-1(x) = ( 4x +3 -3)/4 = x

b) g(x) = 5x/( x- 2)

y = 5x/(x-2)

x = 5y/( y -2)

xy -2x = 5y ; y( x-5) = 2x

y = 2x/( x-5)

f^-1(x) = 2x/(x-5)

To prove f^-1(x) is the inverse of f(x):

f^-1(x) = 2(5x/(X-2) / ( 5x/(x-2) -5)

= x

c) h(x) = 2sqrt(x+1) +3

y = 2sqrt(x+1) +3

x = 2sqrt(y +1) +3

(x-3) = 2sqrt(y+1)

square both sides:

(x -3)^2 = 4(y+1)

y = (x-3)^2/4 -1

f^-1(x) = (x-3)^2/4 -1

To prove f^-1(x) is the inverse of f(x):

f^-1(x) = (2sqrt(x+1) +3-3)^2/4 -1

= (2sqrt(x+1))^2/4 -1

= x