Question

If f(x)=3x-2 and g(x)=x^(2)+1 i. determine f(g(x)) and the domain of f(g(x)) hence find f(g(2))

Answer 1

To find f(g(x)), substitute g(x) into f(x) to get f(g(x)) = 3x² + 1. The domain of f(g(x)) is all real numbers since g(x) is defined everywhere. Substituting 2 into f(g(x)), we find that f(g(2)) = 13.

Step-by-step explanation:

The student is asking how to compose two functions, specifically the function f(x) = 3x - 2 with the function g(x) = x² + 1, and then how to find the domain of this composite function.

To find f(g(x)), we need to substitute g(x) into f(x):

1. First, we take g(x) which is x² + 1.
2. Second, we substitute x² + 1 into f(x) in place of x, giving us f(g(x)) = 3(x² + 1) - 2.
3. Simplify the expression to get f(g(x)) = 3x² + 3 - 2, which simplifies further to f(g(x)) = 3x² + 1.

The domain of f(g(x)) consists of all the values that x can take such that g(x) is defined since f(x) is defined for all real numbers.

Since g(x) is a quadratic polynomial, it is defined for all real numbers, so the domain of f(g(x)) is the set of all real numbers, (-∞, ∞).

To find f(g(2)), we substitue 2 into f(g(x)):

1. f(g(2)) = 3(2²) + 1.
2. Calculate the square of 2, which is 4, so we have f(g(2)) = 3× 4 + 1.
3. Multiplying 3 by 4, we get 12, and then we add 1, yielding f(g(2)) = 13.