Suppose H(X)=(6x-5)^3
Find two functions f and g such that (f•g)(x)=H(x)
Neither function can be the parent/ identity function
f(x) = x^3
g(x) = 6x-5
h(x) = f(g(x)) = g^3 = (6x-5)^3
Well, I'm glad you asked! Since we can't use the parent function, I'll unleash my creativity and come up with two functions that will do the trick – let's call them "Silly Square Root" and "Cheerful Cubic":
Function f(x) = √(2x + 3)
Function g(x) = (x - 2)^2
Now, let's see if they can pull off the magic trick of forming (f•g)(x) = H(x):
(f•g)(x) = f(g(x)) = f((x - 2)^2)
= √(2((x - 2)^2) + 3)
= √(2(x^2 - 4x + 4) + 3)
= √(2x^2 - 8x + 11)
Ah, so close! It seems our combination of Silly Square Root and Cheerful Cubic didn't quite give us H(x) = (6x - 5)^3. But hey, at least we had some fun along the way, right?
To find two functions f and g such that (f•g)(x) equals H(x) = (6x - 5)^3 and neither function can be the parent/identity function, we can use the method of composition of functions.
Let's start by assuming f(x) = (a^3) and g(x) = (bx - c), where a, b, and c are constants to be determined.
Now, let's substitute these functions into the composition (f•g)(x). We have:
(f•g)(x) = f(g(x)) = f(bx - c) = (a^3)((bx - c)^3)
We want this expression to be equivalent to H(x), so we have:
(a^3)((bx - c)^3) = (6x - 5)^3
Expanding both sides of the equation, we get:
a^3(bx - c)^3 = (6x - 5)^3
a^3(b^3x^3 - 3b^2cx^2 + 3bc^2x - c^3) = (6x - 5)^3
Matching the corresponding coefficients on both sides, we have the following equations:
a^3b^3 = 216 ---(1)
-3a^3b^2c = -270 ---(2)
3a^3bc^2 = 90 ---(3)
-a^3c^3 = -125 ---(4)
Solving these equations simultaneously will give us the values of a, b, and c.
From equation (1), we can find b:
b = ∛(216/a^3)
Substituting this value of b into equation (2), we can solve for c:
-3a^3(∛(216/a^3))^2c = -270
-3a^3 * 6^2c = -270
-3a^3 * 36c = -270
a^3c = 7.5
Substituting the value of c = 7.5/a^3 into equation (3), we can solve for a:
3a^3b(7.5/a^3)^2 = 90
3a^3b(7.5^2/a^6) = 90
3 * 7.5^2 * b/a^3 = 90
7.5^2 * b/a = 10
b = 10a/7.5^2
Now substituting the value of b in terms of a back into equation (1), we have:
(10a/7.5^2)^3 = 216
(10^3 * a^3)/(7.5^6) = 216
1000a^3 * 1/(7.5^6) = 216
1000a^3 = 216 * 7.5^6
a^3 = (216 * 7.5^6)/1000
a = ∛((216 * 7.5^6)/1000)
So, we have found the values of a, b, and c. Now, substituting these values back into our assumed functions:
f(x) = (a^3) = (∛((216 * 7.5^6)/1000))^3
g(x) = (bx - c) = ((10a/7.5^2)x - 7.5/a^3)
Therefore, the two functions f and g such that (f•g)(x) = H(x) = (6x - 5)^3 and neither function can be the parent/identity function are:
f(x) = (∛((216 * 7.5^6)/1000))^3
g(x) = ((10(∛((216 * 7.5^6)/1000)))/7.5^2)x - (7.5/(∛((216 * 7.5^6)/1000)))^3
To find two functions f and g such that (f•g)(x) = H(x), where H(x) = (6x-5)^3, we need to decompose H(x) into the composition of two functions.
Let's start by rewriting H(x) as H(x) = f(g(x)).
Let's assume that f(x) = (x - a)^3 and g(x) = (bx + c), where a, b, and c are constants.
Now we can substitute these values into the equation:
(f • g)(x) = H(x)
(f • g)(x) = (f(g(x)))
Substituting f(x) and g(x):
(f • g)(x) = ((g(x) - a)^3)
(f • g)(x) = (((bx + c) - a)^3)
Expanding the expression:
(f • g)(x) = ((bx + c - a)(bx + c - a)(bx + c - a))
Now we need to find the values of a, b, and c that will allow us to rewrite H(x) as a composition of f and g.
To do this, we can compare the expanded expression to the original H(x):
((bx + c - a)(bx + c - a)(bx + c - a)) = (6x - 5)^3
From this comparison, we can see that:
bx + c - a = 6x - 5
We can solve this equation to find the values of a, b, and c:
bx + c - a = 6x - 5
bx + c = 6x - 5 + a
Comparing the coefficients:
b = 6
c = a - 5
So, we have found two functions f(x) and g(x) that satisfy the condition:
f(x) = (x - a)^3
g(x) = (6x + (a-5))
Therefore, (f • g)(x) = H(x) = (6x-5)^3