Question

Let f(x) = g(h(x)) = (√2x+5)/2x. Find two sets of possible formulas for g(x)

and h(x), assuming that g(x) does not equal x and h(x) does not equal x. There may be more than one possible answer.

Answer 1

To find two sets of possible formulas for g(x) and h(x), we can start by examining the given function f(x) = g(h(x)) = (√2x+5)/2x.

One possible set of formulas for g(x) and h(x) is:

g(x) = √x
h(x) = 2x + 5

Substituting these formulas into f(x) = g(h(x)), we get:

f(x) = g(h(x)) = g(2x + 5) = √(2x + 5)

Another possible set of formulas for g(x) and h(x) is:

g(x) = 1/x
h(x) = (√2x + 5)^2/4

Substituting these formulas into f(x) = g(h(x)), we get:

f(x) = g(h(x)) = g((√2x + 5)^2/4) = 4/(2x + 5)

So, we have two possible sets of formulas for g(x) and h(x):

g(x) = √x, h(x) = 2x + 5
g(x) = 1/x, h(x) = (√2x + 5)^2/4

It's worth noting that there may be other possible combinations of g(x) and h(x) that satisfy the given conditions, but these are two examples of such combinations.