Given (f+g)(x)=10-3x and (f-g)(x)=5x-14, find f(x) and g(x).
I'm not sure how to go about this.
To find the functions f(x) and g(x) given the expressions (f+g)(x) and (f-g)(x), we can solve a system of equations using the given information.
Let's solve for f(x) and g(x) step by step:
Step 1: Expand the expressions (f+g)(x) and (f-g)(x).
(f+g)(x) = 10 - 3x
(f-g)(x) = 5x - 14
Step 2: Set up a system of equations using the expanded expressions.
f(x) + g(x) = 10 - 3x ... Equation 1
f(x) - g(x) = 5x - 14 ... Equation 2
Step 3: Solve the system of equations.
To solve this system, we can use the method of elimination. We need to eliminate one variable, either f(x) or g(x), by adding or subtracting both equations. Let's eliminate g(x) by adding the equations together.
(f(x) + g(x)) + (f(x) - g(x)) = (10 - 3x) + (5x - 14)
Simplifying the equation, we get:
2f(x) = 15 - 3x
Divide both sides of the equation by 2:
f(x) = (15 - 3x)/2
Step 4: Substitute f(x) back into either Equation 1 or Equation 2 to solve for g(x). Let's substitute it back into Equation 1:
(15 - 3x)/2 + g(x) = 10 - 3x
To simplify the equation, multiply both sides by 2:
15 - 3x + 2g(x) = 20 - 6x
Now, isolate g(x) by rearranging the equation:
2g(x) = -5 - 3x
Divide both sides by 2:
g(x) = (-5 - 3x)/2
So, the functions f(x) and g(x) are:
f(x) = (15 - 3x)/2
g(x) = (-5 - 3x)/2
(f+g)+(f-g) = 2f
so,
2f(x) = (10-3x)+(5x-14) = 2x-4
f(x) = x-2
You can find g(x) in a similar way.
So, g-g=0
wouldn't that get me g(x)=0?
come on; this is Algebra I
(f+g)-(f-g) = 2g
2g = (10-3x)-(5x-14) = -8x+24
g(x) = -4x+12
or, now that you know f,
f+g-f = g
so,
g(x) = (10-3x)-(x-2) = 12-4x