How would you find the inverse of these two functions: f(x)=4/(x-4)^2 and
f(x)=x^2-8x+16
could you please show me step by step i have been trying to figure it out for hours.
first one:
y = 4/(x-4)^2
inverse is
x = 4/(y-4)^2
solving this for y .....
(y-4)^2 = 4/x
y - 4 = ± 2/√x
y = 4 ± 2/√x
2nd .....
inverse is
x = y^2 - 8y + 16
y^2 - 8y + 16 - x = 0
use the quadratic equation
where a=1, b=-8 and c = 16-x to solve for y
thanks i got the first one now!
for the second one I used the quadratic eqn and got 4 but that is not the answer?
y = (8 ± √(64-4(1)(16-x))/2
= (8 ± √(4x) )/2
= (8 ± 2√x)/2
= 4 ± √x
check: let x = 3 into the first
y = 9 - 24 + 16 = 1
put x = 1 into the second:
y = 4 ± √1 = 5 or 3
notice the the inverse of each of the equations is not a function.
To find the inverse of a function, you need to follow these steps:
Step 1: Replace the function notation f(x) with y.
For the first function, replace f(x) with y:
y = 4/(x-4)^2
For the second function, replace f(x) with y:
y = x^2 - 8x + 16
Step 2: Swap the x and y variables.
In this step, you interchange the x and y variables in the equation. So, the equation becomes:
For the first function:
x = 4/(y-4)^2
For the second function:
x = y^2 - 8y + 16
Step 3: Solve the equation for y.
Now, you need to solve the equation obtained in Step 2 for y. The goal is to isolate y on one side of the equation.
For the first function:
x = 4/(y-4)^2
To solve for y, start by multiplying both sides of the equation by (y-4)^2:
x(y-4)^2 = 4
Expand the square:
xy^2 - 8xy + 16x - 16 = 4
Rearrange the terms:
xy^2 - 8xy + 16x - 20 = 0
Now, this is a quadratic equation in y. To solve, you can use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
Let's identify a, b, and c from the equation:
a = x
b = -8x
c = 16x - 20
Substitute these values into the quadratic formula and simplify to get the inverse of the first function.
For the second function:
x = y^2 - 8y + 16
To solve for y, rearrange the equation:
y^2 - 8y + (16 - x) = 0
Again, this is a quadratic equation in y. Apply the quadratic formula with a, b, and c coefficients identified from the equation to find the inverse of the second function.
By following these steps, you can find the inverse of both functions.