find an expression for F(x) and state the domain of F
when F^-1(x)= 1-5x/2x for 0 < x <= 2
First of all, I will assume you meant:
F^-1(x) = (1-5x)/(2x), or else I would have just reduced to F^-1(x) = 1 - 5/2 = -3/2 , a constant
The inverse of the inverse of a function would be the function, that is,
the inverse of F^-1(x) would be F(x)
so we just need the inverse of y = (1-5x)/(2x)
interchange the x and y's
x = (1-5y)/(2y)
2xy = 1 - 5y
2xy + 5y = 1
y(2x + 5) = 1
y = 1/(2x+5)
F(x) = 1/(2x + 5)
check: let x = 5
F^-1(5) = (1 - 25)/10 = -2.4
F(-2.4) = 1/(-4.8 + 5) = 1/(.2) = 5
It is highly likely that my answer is correct
Just saw the domain part ... 0 < x ≤ 2
F^-1(0) = undefined, but F^-1(x) ----> negative infinity
F^-1(2) = (1-10)/4 = -9/4
so the domain of F(x) is
- infinity < x < -9/4
To find an expression for F(x) and state the domain of F, we need to find the inverse function of F^-1(x) and then solve for x.
Given: F^-1(x) = 1 - 5x / 2x for 0 < x <= 2
Step 1: Solve for x in terms of F^-1(x)
1 - 5x / 2x = F^-1(x)
Multiply both sides by 2x:
2x - 5x = 2x * F^-1(x)
-3x = 2x * F^-1(x)
Divide both sides by 2x:
-3 = F^-1(x)
Now, we have the inverse function F^-1(x) = -3.
Step 2: Find F(x) by finding the inverse of F^-1(x):
Since F^-1(x) = -3, the inverse of this function is x = -3.
So, F(x) = -3.
Step 3: Determine the domain of F
The domain of F is the set of all possible input values for F(x). In this case, the inverse function F^-1(x) was only defined for 0 < x <= 2. Since F(x) = -3, there are no restrictions on the domain of F. Therefore, the domain of F is all real numbers, (-∞, +∞).
To find an expression for F(x), we need to find the inverse function of F^-1(x) and then solve for x.
Given the inverse function F^-1(x) = (1 - 5x) / (2x) for 0 < x ≤ 2, we can find the expression for F(x) by simply interchanging x and F(x). Let's solve for x:
F^-1(x) = (1 - 5x) / (2x)
To interchange x and F(x), we can rewrite it as:
x = (1 - 5F(x)) / (2F(x))
Now, let's solve for F(x). First, let's get rid of the denominator by multiplying both sides by 2F(x):
2xF(x) = 1 - 5F(x)
Next, let's isolate F(x) by collecting all terms with F(x) on one side:
2xF(x) + 5F(x) = 1
(2x + 5)F(x) = 1
Finally, we can divide both sides by (2x + 5) to solve for F(x):
F(x) = 1 / (2x + 5)
Therefore, the expression for F(x) is F(x) = 1 / (2x + 5).
Now, let's determine the domain of F. In the given problem, the inverse function has a restriction of 0 < x ≤ 2. Since the inverse function and the original function are related, we need to check if the original function F(x) has any additional restrictions.
In this case, the denominator of F(x) is 2x + 5, and we know that division by 0 is undefined. Therefore, we need to ensure that 2x + 5 ≠ 0.
To find the domain, solve 2x + 5 ≠ 0 for x:
2x + 5 ≠ 0
2x ≠ -5
x ≠ -5/2
Thus, the domain of F is all real numbers except x = -5/2. Therefore, the domain of F is (-∞, -5/2) ∪ (-5/2, ∞).