
Let f(x) = √(x  4) + 3.
a) Find the
inverse of f.
b) Find the range of f^{ 1}.

Let h(x) = (x  1) / (x + 3).
a) Find the inverse of h.
b) Find the range of h.

Let f(x) = (x  1)/(x + 5) and g(x) = 1/(x + 3).
a) Find the
composite function (f _{o} g)(x).
b) Find the domain of f _{o} g.

Function f is a function with inverse f^{ 1}. Function h is defined by h(x) = f(x) + k where k is a constant. Express the inverse function of h in terms of f^{ 1} and k.

Function f is a function with inverse f^{ 1}. Function h is defined by h(x) = A*f(x  h) + k where A, k and h are constants. Express the inverse function of h in terms of f^{ 1}, A, k and h.

The graphs of functions f and g are shown below.
a) Use the graph to find (f _{o} g)(4)
b) Use the graph to find (g _{o} f)(1)
.
.

Functions f and h are defined by the tables
x 
3 
2 
1 
0 
1 
2 
3 
f(x) 
6 
4 
2 
1 
2 
6 
16 
x 
0 
1 
2 
3 
4 
5 
6 
h(x) 
1 
2 
5 
10 
17 
26 
37 
Use the values in the tables to find
a) (f _{o} h)(1)
b) (f _{o} f)(0)
c) (f _{o} h)(5)
d) (f _{o} h^{1})(5)
e) (h _{o} f^{1})(6)
Answers to the Above Questions


f^{ 1}(x) = (x  3)^{3} + 4 , x≥3
 [4 , +infinity) : it is the domain of f


h^{ 1}(x) = (3x  1) / (x + 1)

(infinity , 1) U (1 , +infinity) : it is the domain of h^{1}


(f _{o} g)(x) = (x + 2) / (5x + 16)

domain of the composite function f _{o} g : (infinity , 16/5) U (16/5 , 3) U (3 , +infinity)

h^{ 1}(x) = f^{ 1}(x  k)

h^{ 1}(x) = f^{ 1}((x  k) / A) + h


(f _{o} g)(4) = f(g(4)) = f(2) = 2

(g _{o} f)(1) = g(f(1) = g(3) = 1


(f _{o} h)(1) = f(h(1)) = f(2) = 6

(f _{o} f)(0) = f(f(0)) = f(1) = 2

(f _{o} h)(5) = f(h(5) = f(26) = undefined

(f _{o} h^{1})(5) = f(h^{1}(5)) = f(2) = 6

(h _{o} f^{1})(6) = h(f^{1}(6)) = h(2) = 5
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