If f (x) = x² + 3 and g (x) = 3x - 1 then find the following:
1. (f/g)(x)
2. (f o g)(x)
3. (f o g)(1)
4. (g o f)(x)
1.
(f/g)(x) = (x^2 + 3)/(3x-1)
2.
(fog)(x) = f(g(x))
= (3x-1)^2 + 3
3.
(fog)(1) = 2^2-3 = 1
4.
(gof)(x) = 3(x^2 + 3) - 1
To find the given values, we will use function composition and perform the necessary calculations step by step.
1. To find (f/g)(x), we need to divide f(x) by g(x). It can be represented as:
(f/g)(x) = f(x) / g(x)
Substitute the given functions f(x) = x² + 3 and g(x) = 3x - 1:
(f/g)(x) = (x² + 3) / (3x - 1)
2. To find (f o g)(x), we need to first evaluate g(x) and then substitute it into f(x).
(f o g)(x) = f(g(x))
Substitute g(x) = 3x - 1 into f(x) = x² + 3:
(f o g)(x) = f(3x - 1) = (3x - 1)² + 3 = 9x² - 6x + 1 + 3 = 9x² - 6x + 4
3. To find (f o g)(1), substitute x = 1 into (f o g)(x):
(f o g)(1) = (f o g)(x) evaluated at x = 1
Substitute x = 1 into (f o g)(x):
(f o g)(1) = 9(1)² - 6(1) + 4 = 9 - 6 + 4 = 7
4. To find (g o f)(x), we need to first evaluate f(x) and then substitute it into g(x).
(g o f)(x) = g(f(x))
Substitute f(x) = x² + 3 into g(x) = 3x - 1:
(g o f)(x) = g(x² + 3) = 3(x² + 3) - 1 = 3x² + 9 - 1 = 3x² + 8
These are the values obtained for the given expressions:
1. (f/g)(x) = (x² + 3) / (3x - 1)
2. (f o g)(x) = 9x² - 6x + 4
3. (f o g)(1) = 7
4. (g o f)(x) = 3x² + 8