Question

If f(x) = 2x + 1 and g(x) = x2, find the values of x for which gf(x) = fg(x).

Answer 1

WRONG ANSWER

I did not read fg(x) and gf(x) but only g(x) and f(x)!!!!

Answer:

x₁= 1-√2  and x₂=1+√2

Step-by-step explanation:

you need to find where the system of the two given equation has solutions

one equation is linear, the other one is a parabolic equation, there will be at most 2 points where they cross path (but there could be even 1 point or even none)

you need to find where 2x+1 = x^2  or  x^2-2x -1 = 0

using the quadratic formula x=(−b±√b2−4ac)/2a

a=1 b=-2 c=-1

you will get two solutions x₁= 1-√2  and x₂=1+√2

Answer 2

Answer:

x = 0, x = - 2

Step-by-step explanation:

Find g(f(x)) and f(g(x))

g(f(x)) = g(2x + 1) = (2x + 1)²

f(g(x)) = f(x²) = 2x² + 1

Thus

(2x + 1)² = 2x² + 1 ← expand left side using FOIL

4x² + 4x + 1 = 2x² + 1 ← subtract 2x² + 1 from both sides

2x² + 4x = 0 ← in standard form

2x(x + 2) = 0 ← in factored form

Equate each factor to zero and solve for x

2x = 0 ⇒ x = 0

x + 2 = 0 ⇒ x = - 2