Could anyone, please, help me solve this?
Let f be a function with the domain the set of all real numbers except 0 and 1 ,that satifies the equation :
2xf(x) - f[(x-1)/x] = 20
Find f(5/4)
note that (x-1)/x = 5/4 if x = -4
Now you have
2(5/4)f(5/4) - f(1/5) = 20
2(1/5)f(1/5) - f(-4) = 20
2(-4)f(-4) - f(5/4) = 20
Now just solve for f(5/4)
Actually I didn't read it thoroughly and quickly replied you. Then I left the site and kept trying to solve it. I got the answers. Now I just came back and saw your replies, which is exactly the same method.
Thank you for your help.
-8f(-4)-f(5/4)=20
or f(5/4)= -8f(-4)-20
Now what to do with f(-4)? How to find it?
come on. Didn't you read what I gave you? Suppose you let
x = f(5/4)
y = f(1/5)
z = f(-4)
Now you just have to find x when you have
5/2 x - y = 20
2/5 y - z = 20
-8z - x = 20
Three equations in three unknowns.
To find the value of f(5/4), we need to solve the given equation for f(x) first.
Let's simplify the equation step by step:
1. Start with the given equation: 2xf(x) - f[(x-1)/x] = 20
2. Substitute x = 5/4 into the equation: 2(5/4)f(5/4) - f[(5/4-1)/(5/4)] = 20
This simplifies to: (5/2)f(5/4) - f(1/4) = 20
3. Now, we need to express f(1/4) in terms of f(5/4). To do this, we can use the given equation.
4. Substitute x = 1/4 into the original equation: 2(1/4)f(1/4) - f[(1/4-1)/(1/4)] = 20
Simplifying this, we get: (1/2)f(1/4) - f(-3/4) = 20
5. Rearrange the equation to solve for f(1/4): (1/2)f(1/4) = 20 + f(-3/4)
Now, we have an expression for f(1/4) in terms of f(-3/4).
6. Substitute this expression into step 2 equation: (5/2)f(5/4) - (20 + f(-3/4)) = 20
Simplify further: (5/2)f(5/4) - f(-3/4) = 40
7. Now, we have two equations:
- (5/2)f(5/4) - f(-3/4) = 40
- (1/2)f(1/4) = 20 + f(-3/4)
8. Solve these two equations simultaneously to find the values of f(5/4) and f(-3/4).
Unfortunately, we can't proceed further without more information or constraints about the function f. The given equation alone is not sufficient to determine the specific function values at 5/4 and -3/4.