Algebra and identifying 2 functions (fog)(x)=H(x)

Question

Suppose H(x)= square root (5 - 4x)

Find two functions f and g that (fog)(x)=H(x)

Neither function can be the identity function.

f(x)= ?

g(x)= ?

Answer 1

The method below will ALWAYS work, get you credit on an exam but is not what your teacher wants and defeats the purpose of the problem.

Let f(x) = x and g(x) = H(x) OR f(x) = H(x) and g(x) = x.

Answer 2

Hi Olivia H.,

With f(g(x)) = H(x) = √(5 - 4x), we need g(x) to plug into f(x) to get H(x).

If we let f(x) = √(x), what do we need to plug into x to get √(5 - 4x)?

We need g(x) = 5 - 4x, to plug into f(x):

f(x) = √(x);

f(g(x)) = √(5 - 4x) = H(x).

I hope this helps, Joe.