(g o f)(x)=12x²+16 x-3 ,f(x)=3x²+4x-3, find g(x)
(g o f)(x)
is the same as f(g(x))
Observing that
12x^2 + 16x - 3
= 4(3x^2 + 4x) - 3
let g(x) = 4x
then f(g(x))
= 3(4x)^2 + 4(4x) - 3
= 12x^2 + 16x - 3 , which is what we have, so
g(x) = 4x
Hmmm. I thought (g o f)(x) was g(f(x)).
If that is so,
12x^2 + 16x - 3 = 4(3x^2 + 4x - 3) + 9
= 4f+9
so g(x) = 4x+9
To find g(x), we need to perform function composition.
The composition of two functions f(x) and g(x) is denoted as (g o f)(x). This means that we first evaluate the function f(x) and then use its output as the input for g(x).
Given that (g o f)(x) = 12x² + 16x - 3 and f(x) = 3x² + 4x - 3, we can substitute the expression for f(x) into the composition formula:
(g o f)(x) = 12x² + 16x - 3 = g(f(x))
Now, we need to solve for g(x). To isolate g(x), we'll need to "undo" the operations performed by f(x) on x.
First, we'll substitute f(x) back into the composition equation:
12x² + 16x - 3 = g(3x² + 4x - 3)
To solve for g(x), we equate the expression on the right side of the equation to g(x):
g(3x² + 4x - 3) = g(x)
Now, we can find g(x) by comparing the terms on both sides of the equation:
g(x) = 12x² + 16x - 3
Therefore, g(x) = 12x² + 16x - 3.