f(x) = 2x+4 and g(x)= x2-1
find (f * g)(4)
find (g * f)(x)
find f^(-1)(x)
To find the composition of two functions, denoted by (f * g)(x), we substitute the expression for g(x) into f(x), and then evaluate the resulting expression.
Given f(x) = 2x + 4 and g(x) = x^2 - 1, we can calculate (f * g)(4) using the following steps:
Step 1: Substitute the expression for g(x) into f(x):
(f * g)(x) = f(g(x))
Step 2: Substitute x with 4 in g(x):
g(4) = 4^2 - 1 = 16 - 1 = 15
Step 3: Substitute the result from Step 2 into f(x):
(f * g)(4) = f(g(4)) = f(15)
Step 4: Substitute x with 15 in f(x):
f(15) = 2(15) + 4 = 30 + 4 = 34
Therefore, (f * g)(4) = 34.
Next, let's find (g * f)(x):
Step 1: Substitute the expression for f(x) into g(x):
(g * f)(x) = g(f(x))
Step 2: Substitute x with the expression for f(x):
g(f(x)) = g(2x + 4)
Step 3: Substitute the result from Step 2 into g(x):
(g * f)(x) = g(f(x)) = (2x + 4)^2 - 1
Therefore, (g * f)(x) = (2x + 4)^2 - 1.
Now, moving on to finding the inverse of f(x), denoted as f^(-1)(x):
Step 1: Replace f(x) with y:
y = 2x + 4
Step 2: Swap x and y:
x = 2y + 4
Step 3: Solve for y:
x - 4 = 2y
(x - 4)/2 = y
Therefore, the inverse of f(x) is f^(-1)(x) = (x - 4)/2.