Answer:
a = -1, b = 10
Step-by-step explanation:
Given the function h(x) = (x – 1)³ + 10 so that h(x) = (g°f)(x). If f(x) = x + a and g(x) = x³ + b, then;
g(f(X)) = g(x+a).
To get g(x+a), we will have to replace the variable x with x+a in g(x) as shown;
g(x+a) = (x+a)³ + b
g(f(x)) = (x+a)³ + b
Since h(x)= g(f(x)) = (x – 1)³ + 10
g(f(x)) = (x+a)³ + b = (x – 1)³ + 10
Hence (x+a)³ + b = (x – 1)³ + 10
On comparing the coefficients to get the value of a and b;
(x+a)³ = (x-1)³
Take the cube root of both sides
∛(x+a)³ = ∛(x-1)³
x+a = x-1
x+a-x = -1
a = -1
Also on comparing, b = 10
Hence the values of a and b would make the composition true are -1 and 10 respectively.
