Question

For the real-valued functions fx=x²+5 and gx=x²+1, find the composition for g.

Answer 1

The composition g(f(x)) for the functions f(x) = x² + 5 and g(x) = x² + 1 is found by substituting f(x) into g(x), resulting in g(f(x)) = (x² + 5)² + 1, which simplifies to x´ + 10x² + 26.

Step-by-step explanation:

To find the composition of the functions f(x) = x² + 5 and g(x) = x² + 1, denoted as g(f(x)), we start by substituting the function f(x) into g(x).

First, we write down f(x):

f(x) = x² + 5

Now, we replace every instance of 'x' in g(x) with f(x):

g(f(x)) = (x² + 5)² + 1

Then, we expand the squared term:

1. (x² + 5)(x² + 5) = x´ + 10x² + 25
2. Now add 1 to the result: x´ + 10x² + 25 + 1
3. The final composition of the function g(f(x)) is: x´ + 10x² + 26

This represents the composition of g(x) after f(x) has been applied to it, which we write as g(f(x)).