Let f(x)=x+2 Ix-3I and g(x)=3x-1
Find all solutions to the equation f(g(x))=-4x
So this is what I did...
(3x-1)+2 I(3x-1)-3I=-4x
(3x-1)+2 I3x-4I=-4x
This is where I got stuck. What do I do with the absolute value? and this part?(3x-1)+2=3x+1 ??
Replace |x-3| with x-3 and solve. Accept only solutions for which x>3
Then replace |x-3| with 3-x and solve again. Accept only solutions for which x<3
Solve it in regions:
a) when 3x>4 or x>4/3
then 3x-1+6x-8=-4x
13x=9
x=9/13 check that. IF so, then not a solution, because x is not>4/3
b) when 3x<4 or x<4/3
then 3x-1-6x+2=-4x
x=1 which is true, because x<4/3
check my figuring.
Thanks you! I get it now!
To solve the equation f(g(x)) = -4x, let's break it down step by step.
First, we need to find g(x) by substituting g(x) = 3x - 1 into f(x):
f(g(x)) = f(3x - 1) = (3x - 1) + 2 |3x - 1 - 3|
Next, let's simplify the absolute value expression:
|3x - 1 - 3| = |3x - 4|
Remember that the absolute value of a number is its distance from zero. In this case, we want to find the value inside the absolute value that makes it equal to zero or positive.
For 3x - 4, we have two cases to consider:
1. When 3x - 4 ≥ 0:
This means that 3x - 4 is greater than or equal to zero. Solving for x:
3x - 4 ≥ 0
3x ≥ 4
x ≥ 4/3
2. When 3x - 4 < 0:
This implies that 3x - 4 is negative. Solving for x:
3x - 4 < 0
3x < 4
x < 4/3
So, we have two cases for the absolute value expression:
1. When 3x - 4 ≥ 0, the absolute value is 3x - 4.
2. When 3x - 4 < 0, the absolute value is -(3x - 4), which can be written as -3x + 4.
Now, let's go back to the equation f(g(x)) = -4x:
(3x - 1) + 2*(3x - 4) = -4x
Simplifying the equation further, we get:
3x - 1 + 6x - 8 = -4x
Combining like terms, we have:
9x - 9 = -4x
Add 4x to both sides:
9x + 4x - 9 = 0
Simplify:
13x - 9 = 0
Now, add 9 to both sides:
13x = 9
Finally, divide both sides by 13:
x = 9/13
So, the solution to the equation f(g(x)) = -4x is x = 9/13.