Find a formula for the inverse of the function. f(x) = (1 + 9x) / (8 - 4x)
let y = (1+9x)/(8-4x)
for inverse, interchange the x and y variables
x = (1+9y)/(8-4y)
now solve this for y .....
8x - 4xy = 1 + 9y
8x - 1 = 9y + 4xy
y(9+4x) = 8x-1
y = (8x-1)/(9+4x)
or
f^-1 (x) = (8x-1)/(9+4x)
check with some arbitrary x
say x = -1
f(-1) = (1-9)/(8+4) = -8/12 = -2/3
f^-1 (-2/3)
= (8(-2/3) - 1)/(9 + 4(-2/3))
= (-19/3) / (19/3)
= -1 , as expected
"more than likely" my answer is correct.
Oh, finding the inverse of a function, eh? Let me juggle some numbers and see if I can come up with a formula for you!
So, we start with the equation f(x) = (1 + 9x) / (8 - 4x). To find the inverse, we need to swap the x and y variables. So, let's call the inverse function g(y) instead.
Step 1: Swap the x and y variables:
x = (1 + 9y) / (8 - 4y)
Step 2: Solve for y:
8 - 4y = 1 + 9y
8 - 1 = 9y + 4y
7 = 13y
y = 7/13
Step 3: Swap the y and x variables back:
y = 7/13
And there you have it! The inverse function is g(y) = 7/13. Now, that might not be what you were expecting, but hey, sometimes math can be a real joker, right?
To find the inverse of the function f(x) = (1 + 9x) / (8 - 4x), we can follow these steps:
Step 1: Replace f(x) with y to obtain the equation: y = (1 + 9x) / (8 - 4x).
Step 2: Swap the roles of x and y, so the equation becomes x = (1 + 9y) / (8 - 4y).
Step 3: Solve the equation for y. Begin by cross-multiplying: x(8 - 4y) = 1 + 9y.
Step 4: Expand the equation by distributing the x: 8x - 4xy = 1 + 9y.
Step 5: Rearrange the terms to isolate the y: 9y + 4xy = 8x + 1.
Step 6: Move the 4xy term to the right side of the equation: 9y = 8x + 1 - 4xy.
Step 7: Factor out the common term on the right side: 9y = 1 + x(8 - 4y).
Step 8: Divide both sides of the equation by (8 - 4y): (9y) / (8 - 4y) = (1 + x(8 - 4y)) / (8 - 4y).
Step 9: Cancel out the common term on the right side: (9y) / (8 - 4y) = 1 + x.
Step 10: Subtract 1 from both sides of the equation: (9y) / (8 - 4y) - 1 = x.
Step 11: Multiply both sides of the equation by (8 - 4y) to eliminate the fraction: 9y - (8 - 4y) = x(8 - 4y).
Step 12: Simplify the left side: 9y - 8 + 4y = x(8 - 4y).
Step 13: Combine like terms: 13y - 8 = x(8 - 4y).
Step 14: Divide both sides of the equation by (8 - 4y): (13y - 8) / (8 - 4y) = x.
Therefore, the formula for the inverse of the function f(x) = (1 + 9x) / (8 - 4x) is x = (13y - 8) / (8 - 4y).
To find the inverse of a function, we need to solve for x in terms of y in the equation f(x) = y.
Let's start by replacing f(x) with y in the given function:
y = (1 + 9x) / (8 - 4x)
Next, let's swap the variables x and y in the equation:
x = (1 + 9y) / (8 - 4y)
Our goal is to solve this equation for y. To do that, we'll manipulate the equation step by step:
1. Distribute 4 to both terms in the denominator:
x = (1 + 9y) / 8 - (4y / 8)
2. Simplify the expression:
x = (1 + 9y) / 8 - y / 2
3. Multiply the entire equation by 8 to eliminate the denominator:
8x = 1 + 9y - 4y
4. Combine like terms:
8x = 1 + 5y
5. Subtract 1 from both sides of the equation:
8x - 1 = 5y
6. Divide both sides of the equation by 5:
(8x - 1) / 5 = y
Now that we have y in terms of x, we can replace y with f^(-1)(x) (the inverse function) to obtain the final formula:
f^(-1)(x) = (8x - 1) / 5
Therefore, the inverse of the given function f(x) = (1 + 9x) / (8 - 4x) is f^(-1)(x) = (8x - 1) / 5.