Evaluate the given function for the given x value.
f(x)=x2+1/4X+2,g(X)=x3+1/2 X2+x
find (f+g)(a)
Explain
To evaluate the combined function (f+g)(a), we need to find the sum of the two functions f(x) and g(x) when x is replaced with the given value a.
First, let's find f(x+a):
f(x+a) = (x+a)^2 + (1/4)(x+a) + 2
= x^2 + 2ax + a^2 + (1/4)x + (1/4)a + 2
= x^2 + (2a + 1/4)x + (a^2 + (1/4)a + 2)
Next, let's find g(x+a):
g(x+a) = (x+a)^3 + (1/2)(x+a)^2 + (x+a)
= (x^3 + 3x^2a + 3xa^2 + a^3) + (1/2)(x^2 + 2ax + a^2) + (x + a)
= x^3 + (3a + 1/2)x^2 + (3a^2 + a)x + (a^3 + (1/2)a^2 + a)
Finally, let's find (f+g)(a):
(f+g)(a) = (a)
= (a^2 + (2a + 1/4)a + (a^2 + (1/4)a + 2)) + (a^3 + (3a + 1/2)a^2 + (3a^2 + a)a + (a^3 + (1/2)a^2 + a))
= 2a^2 + (11/4)a + 3
So, the value of (f+g)(a) is 2a^2 + (11/4)a + 3.