Step 1
We know that,
f(x)=4x+2 and g(x)=2x−3
(1) The value of f(g(x)) can be evaluated as
f(g(x))=4(g(x))+2
=4(2x−3)+2
=8x−12+2
=8x−10
=2(4x−5)
(2) We know that, given a function f(x), a new function f'(x)=f(x)+k, where k is a constant, is a vertical shift of the function f(x). If k is positive, the graph will shift up. If k is negative, the graph will shift down.
Hence, the transformation of f(x)=(4x+2)−1 is 1 unit down from the graph of f(x).
(3) We know that, given a function f(x), a new function f'(x)=f(x)+k, where k is a constant, is a vertical shift of the function f(x). If k is positive, the graph will shift up. If k is negative, the graph will shift down.
Hence, the transformation of f(x)=4(x + 3)+2 is 2 unit up from the graph of f(x).
Step 2 (4) Given a function f(x)a new function g(x)=af(x),where a is a constant, is a vertical stretch or vertical compression of the function f(x) takes place as
1.If a>1, then the graph will be stretched.
2.If 0 3.If a<0, then there will be combination of a vertical stretch or compression with a vertical reflection.
Hence, the transformation of g(x) = 2(2x - 3) is that the graph of g(x)=(2x-3) will be stretched.
(5) We have,
f(x)=4(3x)+2
=12x+2
=3(4x+2−2)+2
=3(4x+2)−6+2
=3(4x+2)−4
Hence, the transformation of f(x)=4(3x)+2 is that firstly the graph of f(x)=(4x−2) will be stretched vertically by a factor 3 as and then shift 4 unit down.