To determine the values of "x" that satisfy the equation f(g(x)) = g(f(x)), we first need to find the expressions for f(g(x)) and g(f(x)).
a) Given:
f(x) = 2x + 3
g(x) = x^2
We need to find x such that f(g(x)) = g(f(x)).
f(g(x)) = f(x^2) = 2(x^2) + 3 = 2x^2 + 3
g(f(x)) = g(2x + 3) = (2x + 3)^2 = 4x^2 + 12x + 9
Setting f(g(x)) equal to g(f(x)), we have:
2x^2 + 3 = 4x^2 + 12x + 9
Rearranging the equation:
2x^2 - 4x^2 - 12x = 9 - 3
-2x^2 - 12x = 6
Dividing through by -2:
x^2 + 6x = -3
Adding 3 to both sides:
x^2 + 6x + 3 = 0
This quadratic equation does not factor easily, so we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Using the quadratic formula:
x = (-6 ± √(6^2 - 4(1)(3))) / (2(1))
x = (-6 ± √(36 - 12)) / 2
x = (-6 ± √24) / 2
x = (-6 ± 2√6) / 2
x = -3 ± √6
Therefore, the values of x that satisfy f(g(x)) = g(f(x)) for the given functions are x = -3 + √6 and x = -3 - √6.
b) Given:
f(x) = x^2
g(x) = 2x + 1
We need to find x such that f(g(x)) = g(f(x)).
f(g(x)) = f(2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1
g(f(x)) = g(x^2) = 2(x^2) + 1 = 2x^2 + 1
Setting f(g(x)) equal to g(f(x)), we have:
4x^2 + 4x + 1 = 2x^2 + 1
Subtracting 2x^2 and 1 from both sides:
2x^2 + 4x = 0
Factoring out 2x:
2x(x + 2) = 0
Setting each factor equal to zero:
2x = 0 or x + 2 = 0
x = 0 or x = -2
Therefore, the values of x that satisfy f(g(x)) = g(f(x)) for the given functions are x = 0 and x = -2.
7) For each pair of functions, determine values of “x” such that
f g x g f x .
a)
2
f x x g x x x 2 3 ; 3
b)
; 2 1
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