Find the Inverse Function:
y=4x-x^2
You were given:
y = 4x - x^2
We switch x and y.
x = 4y - y^2
Factor the right side of the equation.
x = y(4 - y)
Next:
xy = 4 - y
Add y on both sides of the equation.
xy + y = 4
Factor on the left side.
y(x + 1) = 4
Isolate y on the left side.
y = 4/(x + 1)
Replace y by f^-1 (x).
f^-1 (x) = 4/(x + 1)
Done!
How did you go from here:
x = y(4 - y)
To here:
Next:
xy = 4 - y
Shouldn't you divide rather than multiply?
Mike,
After doing the question again, I realize that 4x - x^2 is not one-to-one. Hence, it has no inverse function.
Sorry....The answer is: NO INVERSE FUNCTION HERE.
To find the inverse function of y = 4x - x^2, we need to first solve for x in terms of y. Then we can interchange x and y to obtain the inverse function.
Step 1: Start with the given equation: y = 4x - x^2.
Step 2: Replace y with x and x with y: x = 4y - y^2.
Step 3: Rearrange the equation to solve for y: y^2 - 4y + x = 0.
Step 4: This equation is a quadratic equation in terms of y. To solve it, we can use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 1, b = -4, and c = x.
Plugging these values into the formula, we get:
y = (4 ± √((-4)^2 - 4(1)(x))) / (2(1))
= (4 ± √(16 - 4x)) / 2
= (4 ± √(16 - 4x)) / 2
= 2 ± √(4 - x)
Step 5: Now we have two possible values for y: y = 2 + √(4 - x) and y = 2 - √(4 - x).
So, the inverse function of y = 4x - x^2 is:
y = 2 + √(4 - x)
and
y = 2 - √(4 - x)