Find f(x) and g(x) so that the function can be described as y=f(g(x)).
y = |3x-2|
(With explanation too please!)
To find functions f(x) and g(x) such that the given function can be described as y = f(g(x)), we need to break down the given function into two simpler functions.
The given function is y = |3x - 2|. The absolute value function |x| can be defined in terms of a piecewise function:
|x| = x, if x ≥ 0
|x| = -x, if x < 0
So, let's break down the given function into two cases:
Case 1: For x ≥ 0
For x ≥ 0, the given function can be written as:
y = 3x - 2
In this case, g(x) = x, and f(x) = 3x - 2.
Case 2: For x < 0
For x < 0, the given function can be written as:
y = -(3x - 2) = -3x + 2
In this case, g(x) = -x, and f(x) = -3x + 2.
So, the functions f(x) and g(x) that describe the given function y = |3x - 2| as y = f(g(x)) are:
g(x) = x, if x ≥ 0
g(x) = -x, if x < 0
f(x) = 3x - 2, if x ≥ 0
f(x) = -3x + 2, if x < 0
In summary:
g(x) = |x|
f(x) = 3x - 2, if x ≥ 0
f(x) = -3x + 2, if x < 0
That's how we can describe the given function y = |3x - 2| as y = f(g(x)).
Let f(x) = |x|
so, now, y = f(g(x)) = |g(x)|
so, what do you think g(x) ought to be?