The value of f(6 + h) − f(6) is 36 + 6h - g^2 - 2gh - h^2.
Given,
f(x) = 6x − x2
We need to find f(6 + h) − f(6)
We have,
f(x) = 6x − x2
Find f(6 + h)
f(x) = 6x - x^2
Putting x = 6 + h
f(6+h)
= 6(6+h) - (g+h)^2
= 36 + 6h - (g^2 + 2gh + h^2)
= 36 + 6h - g^2 - 2gh - h^2 ________(A)
Find f(6)
f(x) = 6x - x^2
f(6)
= 6(6) - 6^2
= 36 - 36
= 0 ________(B)
Find f(6 + h) − f(6)
From A and B
= 36 + 6h - g^2 - 2gh - h^2 - 0
= 36 + 6h - g^2 - 2gh - h^2
Thus the value of f(6 + h) − f(6) is 36 + 6h - g^2 - 2gh - h^2.
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Parent function: y = f(x) = x^2
given function: y = g(x) = 3 (x+1)^2
=> g(x) = 3 * f(x + 1)^2
So, from the point of view of graphs, the given function, g(x), is the parent function, f(x) = x^2, shifted one unit to the left and amplified by a factor of 3.
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