There's f(x) = (2x+3)/(x+4)
and g(x) = (4x-3)/(2-x)
Solve for x in f(g(x))
I tried this but I could not end up with the answer being x
f(g(x) = [2(4x-3)/(2-x) + 3]/[(4x-3)/(2-x) + 4]
= [(8x-6 + 6-3x)/(2-x)] / [(4x-3 + 8-4x)/(2-x)]
= [5x/(2-x)] / [5/(2-x)]
= [5x/(2-x)] [(2-x)/5]
= 5x/5
= x
thank you so much
welcome
To solve for x in f(g(x)), we need to substitute the function g(x) into f(x) and then simplify the expression. Let's go through the steps together:
Step 1: Start by replacing f(x) with g(x) in the expression for f(g(x)):
f(g(x)) = f((4x-3)/(2-x))
Step 2: Simplify the expression for f(g(x)):
f(g(x)) = [(2((4x-3)/(2-x))) + 3]/(((4x-3)/(2-x)) + 4)
= [(8x-6)/(2-x) + 3]/((4x-3+ (2-x)(4))/(2-x))
Step 3: Next, simplify the numerator:
Numerator = (8x-6 + (2-x)(3(2-x)))/(2-x)
= (8x-6 + (6-3x)(2-x))/(2-x)
= (8x-6 + (12-6x-3x^2+3x))/(2-x)
= (8x-6 -3x^2 +6 -6x +3x)/(2-x)
= (-3x^2 +5x)/(2-x)
Step 4: Simplify the denominator:
Denominator = 4x-3 + (2-x)(4)
= 4x-3 + (8-4x)
= 4x-3 + 8 -4x
= 5
Step 5: Now we can rewrite f(g(x)):
f(g(x)) = (-3x^2 + 5x)/5
Step 6: Set f(g(x)) equal to x and solve for x:
(-3x^2 + 5x)/5 = x
To simplify further, we can multiply both sides of the equation by 5 to eliminate the denominator:
-3x^2 + 5x = 5x
Move all terms to one side of the equation:
-3x^2 - 5x + 5x = 0
Combine like terms:
-3x^2 = 0
Divide both sides by -3:
x^2 = 0
Take the square root of both sides to solve for x:
x = 0
So, the solution to f(g(x)) = x is x = 0.
I hope this clears up any confusion! Let me know if there's anything else I can help with.