Find the composite functions for the given functions. (6 marks)
f(x) = 4x + 1 and g(x) = x2
f(x) = sin x and g(x) = x2 - x + 1
f(x) = 10x and g(x) = log x
To find the composite functions, we need to substitute one function into another function. We will start with the first set of functions.
1. f(x) = 4x + 1 and g(x) = x^2
To find the composite function f(g(x)), we will substitute g(x) into f(x):
f(g(x)) = f(x^2)
Now, let's substitute the value of g(x) into f(x):
f(g(x)) = 4(x^2) + 1
Simplifying this expression, we get:
f(g(x)) = 4x^2 + 1
2. f(x) = sin(x) and g(x) = x^2 - x + 1
To find the composite function f(g(x)), we will substitute g(x) into f(x):
f(g(x)) = f(x^2 - x + 1)
Now, let's substitute the value of g(x) into f(x):
f(g(x)) = sin(x^2 - x + 1)
3. f(x) = 10x and g(x) = log(x)
To find the composite function f(g(x)), we will substitute g(x) into f(x):
f(g(x)) = f(log(x))
Now, let's substitute the value of g(x) into f(x):
f(g(x)) = 10(log(x))
Therefore, the composite functions are:
1. f(g(x)) = 4x^2 + 1
2. f(g(x)) = sin(x^2 - x + 1)
3. f(g(x)) = 10(log(x))