SOLUTION: The function f is such that f(x) = 3x - 2 for x >= 0. The function g is such that g(x) = 2x^2 - 8 for x <= k, where k is a constant. (a) Find the greatest value of k for which

Question 1186585: The function f is such that f(x) = 3x - 2 for x >= 0.
The function g is such that g(x) = 2x^2 - 8 for x <= k, where k is a constant.
(a) Find the greatest value of k for which the composite function fg can be formed.
(b) For the case where k = -3
(i) find the range of fg

(ii) find (fg)^-1(x) and state the domain and the range of (fg)^-1.
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
Given that the (restricted) domain of f(x) = 3x - 2 is [0,), we need to choose the values of for the composition f o g to be possible.

(a) Hence, ===> , or x ∈ (, -2] U [2, ). By virtue of continuity, the greatest value of k for which the composite function f o g can be formed is .

(b) (i) Now . Since the domain of f o g is (, -3], the expression maps (, -3] onto the interval [28, ).
This is the range of (f o g)(x).

(ii) To find , let .
===> ===> . (Choose the negative part since this is an element of (, -3].)
===> after interchanging the places of x and y.

===> . Its domain is the range of f o g, namely [28, ), while its range is the domain of f o g, namely (, -3].

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