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)
To find the value of these composite functions, we follow the order of operations and perform the necessary operations on the given functions. Let's solve each part step by step:
1. To find (f/g)(x), we need to divide f(x) by g(x) and substitute x into the resulting expression.
f(x) = x² + 3
g(x) = 3x - 1
Therefore, (f/g)(x) = (x² + 3)/(3x - 1)
2. To find (f o g)(x), we need to first find g(x), and then substitute it into f(x). This means we need to find f(g(x)).
f(x) = x² + 3
g(x) = 3x - 1
Therefore, (f o g)(x) = f(g(x)) = f(3x - 1)
Substitute 3x - 1 into f(x):
(f o g)(x) = (3x - 1)² + 3
3. To find (f o g)(1), we need to evaluate (f o g)(x) when x = 1.
(f o g)(x) = (3x - 1)² + 3
Substitute x = 1 into (f o g)(x):
(f o g)(1) = (3(1) - 1)² + 3
4. To find (g o f)(x), we need to first find f(x), and then substitute it into g(x). This means we need to find g(f(x)).
f(x) = x² + 3
g(x) = 3x - 1
Therefore, (g o f)(x) = g(f(x)) = g(x² + 3)
Substitute x² + 3 into g(x):
(g o f)(x) = 3(x² + 3) - 1
Now, you can simplify the expressions (f/g)(x), (f o g)(x), and (g o f)(x) further if needed by performing the required operations. To find the specific values of (f o g)(1) and (g o f)(x), substitute the appropriate values into the expressions and calculate the results.