If f(x)=x-3 and g(x)=x^2 -4x+1, find the minimum value of (g o f)(x)

Question

Find the minimum value of (g o f)(x)

Answer 1

Greetings! Lets solve this shall we ?

First, lets find g[f(x)] such that we plug f(x) = x - 3 into g(x) = x²-4x+1 to become

g[f(x)] = (x-3)² - 4(x-3) + 1

= (x²-6x+9) - 4x + 12 + 1

= x² - 10x + 9 + 13

g[f(x)] = x² - 10x + 22

In order to find the minimum value, we must find the vertex of g[f(x)] such that (-b/2a, f(-b/2a)) where a = 1 and b = -10 in this case. Then,

(10/2, f(10/2)) = (5, f(5))

and f(5) = (5)² - 10(5) + 22 = 25 - 50 + 22 = -25 + 22 = -3.

So the vertex is (5, -3)

Thus, the minimum value is the y-value of -3.

I hope this helped!