F(x)=5e^x - 10 / 24e^x + 12
f^(-1)(x)=?
a=?
b=?
To find the inverse of a function, f^(-1)(x), we need to swap the roles of x and y in the original function and solve for y. In this case, we have the function f(x) = (5e^x - 10) / (24e^x + 12).
Step 1: Swap x and y
x = (5e^y - 10) / (24e^y + 12)
Step 2: Solve for y
To solve for y, we start by multiplying both sides of the equation by (24e^y + 12) to eliminate the denominator.
x(24e^y + 12) = 5e^y - 10
Step 3: Distribute and collect like terms
24xe^y + 12x = 5e^y - 10
Step 4: Move the terms with e^y to one side and the constant terms to the other side
24xe^y - 5e^y = -12x - 10
Step 5: Factor out e^y from the terms on the left side
e^y(24x - 5) = -12x - 10
Step 6: Divide both sides by (24x - 5) to isolate e^y
e^y = (-12x - 10) / (24x - 5)
Step 7: Take the natural logarithm (ln) of both sides to solve for y
y = ln((-12x - 10) / (24x - 5))
Therefore, the inverse function is f^(-1)(x) = ln((-12x - 10) / (24x - 5)).
Now, regarding the variables a and b, it seems they are not specified clearly in the given problem. Without proper information about a and b, it is not possible to determine their values or their relationship to the function f(x) = (5e^x - 10) / (24e^x + 12).