Question

Let f(x) = axᵇ and g(x) = cxᵈ, where a, b, c, and d are constants. Determine the necessary and sufficient conditions on the constants a, b, c, and d so that f —¦ g = g —¦ f?

Answer 1

The necessary and sufficient conditions for f(x) = ax^b to be equal to g(x) = cx^d are ab = db and ac^b = ca^d.

Step-by-step explanation:

To determine the necessary and sufficient conditions for f(x) = ax^b to be equal to g(x) = cx^d, we need to find the values of a, b, c, and d that satisfy the equation f(g(x)) = g(f(x)).

Substituting f(x) and g(x) into the equation, we have:

a(c*x^d)^b = c(a*x^b)^d

Simplifying the equation, we get:

ac^bx^bd = ca^dx^db

For the equation to hold true for all values of x, the coefficients on each side of the equation must be equal:

ab = db and ac^b = ca^d

Therefore, the necessary and sufficient conditions for f(x) = ax^b to be equal to g(x) = cx^d are ab = db and ac^b = ca^d.