Suppose H(x)=(sqrt5x+3).
Find two functions f and g such that (fog)(x)= H(x) .
f(x)=
g(x)=
Neither function can be the identity function.
(There may be more than one correct answer.)
g(x) = sqrt(5x)
f(x) = x + 3
Thank you
To find two functions f and g such that (fog)(x) = H(x), we need to consider how the composition of functions works.
Let's start by expressing H(x) = √(5x + 3).
Notice that the inner function g(x) must be such that applying it will result in √(5x + 3).
One possible choice for g(x) could be g(x) = 5x + 3.
Now, let's find the function f(x) such that (fog)(x) = H(x), which means applying f to g(x) will give us the desired output.
Substituting g(x) into the expression, we have f(g(x)) = f(5x + 3).
To eliminate the square root, we need f(x) to undo the square root operation. Therefore, a possible choice for f(x) is f(x) = x^2.
Let's verify if (fog)(x) = H(x) by calculating (fog)(x) and comparing it to H(x):
(fog)(x) = f(g(x)) = f(5x + 3) = (5x + 3)^2.
Expanding the squared term, we get:
(5x + 3)^2 = 25x^2 + 30x + 9.
Comparing this with H(x) = √(5x + 3), we see that (fog)(x) indeed equals H(x).
Therefore, one possible solution is:
f(x) = x^2
g(x) = 5x + 3
To find two functions f and g such that (fog)(x) = H(x), we need to consider the composite function (fog)(x) and try to express it in terms of H(x) = (√5x + 3).
Let's start by writing the composite function (fog)(x) using the variables f and g:
(fog)(x) = f(g(x))
To find f and g, we need to find the individual functions f(x) and g(x) that when composed together, yield H(x).
One possible solution for f and g can be:
f(x) = (√5x) - 3
g(x) = (√x + 3)
Now let's verify that (fog)(x) = H(x):
(fog)(x) = f(g(x))
= f(√x + 3)
= (√5(√x + 3)) - 3
= (√5√x + √5(3)) - 3
= (√5√x + 3√5) - 3
= √5x + 3
As we can see, (fog)(x) = H(x). Therefore, f(x) = (√5x) - 3 and g(x) = (√x + 3) are functions that satisfy the given condition.