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
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.
To determine the values of "x" such that f(g(x)) = g(f(x)), we need to substitute the given functions f(x) and g(x) into the equation and solve for x.
a) For the functions:
f(x) = x^2 + 2x + 3
g(x) = x - 2
Substituting the functions into the equation:
f(g(x)) = g(f(x))
(x - 2)^2 + 2(x - 2) + 3 = (x^2 + 2x + 3)
Expanding and simplifying:
x^2 - 4x + 4 + 2x - 4 + 3 = x^2 + 2x + 3
x^2 - 2x + 3 = x^2 + 2x + 3
Rearranging the terms:
-2x = 2x
Dividing both sides by 2:
x = 0
Therefore, the value of "x" for which f(g(x)) = g(f(x)) is x = 0.
b) The functions are not provided, so we are unable to determine the values of "x" for this case. Please provide the functions f(x) and g(x) for part b) so that we can proceed with the solution.