Question

Consider the functions f(x)=x² −1 and g(x)=x+1. For each operation below, simplify the maximum expression.

a) f−g
b) f+g
c) f or g
d) g or f

Answer 1

The operations involving the functions f(x) = x² − 1 and g(x) = x + 1 yield the following simplifications: f − g results in x² − x − 2, f + g yields x² + x, f ∘ g gives x² + 2x, and g ∘ f simplifies to x².

Step-by-step explanation:

The student has asked to simplify the expressions resulting from the combination of two functions, f(x) and g(x). Here are the simplifications for each operation:

• f − g: Subtraction of functions means we subtract g(x) from f(x). So, we have f(x) − g(x) = x² − 1 − (x + 1) = x² − x − 2.
• f + g: Addition of functions means we add g(x) to f(x). So, f(x) + g(x) = x² − 1 + (x + 1) = x² + x.
• f ∘ g: Composition of functions, (f ∘ g)(x), means we apply g first, then f. So, we get f(g(x)) = f(x + 1) = (x + 1)² − 1 = x² + 2x + 1 − 1 = x² + 2x.
• g ∘ f: Again, the composition of functions, (g ∘ f)(x), means we apply f first, then g. So, g(f(x)) = g(x² − 1) = (x² − 1) + 1 = x².