I have no idea how to do inverse functions...
How would I do this question?
Y=-3x^2 - 18x - 25
well, first of all that is a parabola as you know.
If you invert it, it is likely to have two values of the ordinate for every value of the abscissa. In other words the inverse will not likely really be a function.
However to do what you want to do, use x for y and y for x and then solve for y.
x = -3y^2 - 18 y - 25
3 y^2 + 18 x + 25 = -x
y^2 + 6 y + 25/3 = -x/3
y^2 + 6 y = -x/3 -25/3
y^2 + 6 y + 9 = -x/3 -25/3 + 27/3
(y+3)^2 = -x/3 + 2/3
right there you can see that for a single value of x you get two values of y, one + and one - because the squares are the same.
y+3= + or - sqrt (2/3 - x/3)
Oh and also it is imaginary for (2-x) negative or x > 2
In other words the new inverse curve is open to the left with a vertex at (2,-3)
To find the inverse of a function, you need to follow a few steps:
Step 1: Rewrite the equation by replacing "y" with "x" and "x" with "y".
So, in your case, the equation is: x = -3y^2 - 18y - 25.
Step 2: Rearrange the equation and express "y" in terms of "x". Solve for "y" to find the inverse function.
Let's go through the process step by step:
x = -3y^2 - 18y - 25
Step 3: Swap the variables "x" and "y".
x = -3x^2 - 18x - 25
Step 4: Rearrange the equation to solve for "y".
0 = -3x^2 - 18x - 25 - x
0 = -3x^2 - 19x - 25
Step 5: Use the quadratic formula or factoring to solve for "x".
Using the quadratic formula, where a = -3, b = -19, and c = -25:
x = (-b ± √(b^2 - 4ac)) / 2a
x = (-(-19) ± √((-19)^2 - 4(-3)(-25))) / (2(-3))
Simplify the formula:
x = (19 ± √(361 - 300)) / (-6)
x = (19 ± √(61)) / (-6)
So, the two possible values for "x" are:
x₁ = (19 + √(61)) / (-6)
x₂ = (19 - √(61)) / (-6)
Step 6: Replace "x" with "y" to find the inverse function.
Now we have the values of "x" in terms of "y":
x₁ = (19 + √(61)) / (-6) = f(y)
x₂ = (19 - √(61)) / (-6) = f(y)
Hence, the inverse function of Y = -3x^2 - 18x - 25 is:
f^(-1)(y) = (19 + √(61)) / (-6) or f^(-1)(y) = (19 - √(61)) / (-6)
These are the two possible ways to express the inverse function of the given equation.