Find f(x) and g(x) so that the function can be described as y = f(g(x)). Given y = (10)/(√(2x + 9).
f(x) = 10 , g(x) = √(2x + 9)
f(x) = 10/√(x) , g(x) = 2x + 9
f(x) = 10/x , g(x) = 2x + 9
f(x) = √(2x + 9) , g(x) = 10

Solution:

In this problem y is called a composite function of f(x) and g(x). So in the options given in the problem statement which option will satisfy y = f(g(x)) is to be found out.

Let f(x) be defined as some function then if ‘x’ is replaced by g(x) in f(x) i.e. f(g(x)) then its value will be y. Hence we can write:

y = f(g(x)) --- (1)

We are also given that

y = (10)/(√(2x + 9))

Let us consider each of the cases given under options, one by one. We have to find out the nature of the function f(x) as well as g(x) so that we substitute g(x) for x in f(x) we have:

y = (10)/(√(2x + 9))

(i) Consider f(x) = 10 and g(x) = √(2x + 9)

Here f(x) cannot take g(x) as its inputs.

(ii) Consider f(x) = 10/√(x) , g(x) = 2x + 9

Here we determine that f(g(x)) = (10)/(√(2x + 9))

(iii) Consider f(x) =10/x , g(x) = 2x + 9

Here f(g(x) = 10/(2x+9)

(iv) Consider f(x)= √(2x + 9) , g(x) = 10

f(g(x) = f(10) = √(2(10) + 9) = √29

From the options given in the problem statement the values of f(x) and g(x) that satisfy equation (2) i.e. y = f(g(x)) = (10)/(√(2x + 9))are:

f(x) = 10/√(x) and

g(x) = 1/(2x + 9)

---

Find f(x) and g(x) so that the function can be described as y = f(g(x)). Given y = (10)/(√(2x + 9).

Summary:

The values of f(x) and g(x) that describe y = f(g(x)) (given y = (10)/(√(2x + 9)) are 10 /√(x) and 2x + 9 respectively.