find F(y) and f(y) if Y=-4X+3 and
f(x)=2(e^-2x)u(x)
To find F(y) and f(y), we need to manipulate the given equations and substitute the given variables accordingly.
Given:
Y = -4X + 3 -------------- (Equation 1)
f(x) = 2(e^(-2x))u(x) ---- (Equation 2)
To find F(y), we need to solve Equation 1 for X.
1. Start by isolating X on one side of the equation:
-4X = Y - 3
2. Divide both sides of the equation by -4:
X = (Y - 3)/(-4)
Now we have an expression for X in terms of Y. Next, we can use this to find f(y) by substituting it into Equation 2:
f(y) = 2(e^(-2((Y - 3)/(-4))))u((Y - 3)/(-4))
Simplifying further, we can simplify the exponent and rewrite the step as:
f(y) = 2(e^(1/2(Y - 3)))u((Y - 3)/(-4))
Now we have the expression for f(y) in terms of Y.
In summary:
F(y) = (Y - 3)/(-4)
f(y) = 2(e^(1/2(Y - 3)))u((Y - 3)/(-4))
Please note that u(x) represents the unit step function, which ensures the function is zero for negative x and unity for positive x. It acts as a switch to consider only the positive values.