The function f(x)=2x+8/3x-7 is one to one. Find its inverse, and check your answer.
f^-1(x)=
I haven't one clue how to do this...help me pleaaaase someone!
let y = (2x+8)/(3x-7) , notice the brackets are essential
step1: interchange the x and y variables to form the initial inverse.
x = (2y+8)/(3y-7)
step: solve this new equation for y
3xy - 7x = 2y+8
3xy - 2y = 7x + 8
y(3x - 2) = (7x + 8)
y = (7x + 8)/(3x - 2)
f^-1 (x) = (7x + 8)/(3x - 2)
test with some value of x, say x = 3
f(3) = 14/2 = 7
f^-1 (7) = should get 3 back ..
= (49+8)/(21-2) = 57/19 = 3 , looks ok
To find the inverse of a function, we need to switch the roles of x and y in the original function and solve for y.
In this case, our original function is f(x)=2x+8/3x-7.
Step 1: Replace f(x) with y.
y = 2x + 8/3x - 7
Step 2: Swap x and y.
x = 2y + 8/3y - 7
Step 3: Solve for y.
To eliminate fractions, we can multiply both sides of the equation by the least common denominator of 3y.
3yx = 2y(3y) + (8/3y)(3y) - 7(3y)
3yx = 6y^2 + 8 - 21y
Rearranging the terms:
6y^2 - 3yx - 21y + 8 = 0
Step 4: Use the quadratic formula.
Since we have a quadratic equation, we can use the quadratic formula to solve for y.
The quadratic formula is: y = (-b ± sqrt(b^2 - 4ac)) / 2a
For our equation 6y^2 - 3yx - 21y + 8 = 0, we have:
a = 6
b = -3x
c = -21
Plugging those values into the quadratic formula, we get:
y = (-(-3x) ± sqrt((-3x)^2 - 4(6)(-21))) / (2(6))
y = (3x ± sqrt(9x^2 + 504)) / 12
Therefore, the inverse function of f(x) is:
f^(-1)(x) = (3x ± sqrt(9x^2 + 504)) / 12
To check our answer, we can compose the original function with its inverse and see if we get back x.
f(f^(-1)(x)) = x
If we substitute f^(-1)(x) into the original function, we should get x as the result.
f(f^(-1)(x)) = 2(f^(-1)(x)) + 8 / 3(f^(-1)(x)) - 7
= (2((3x ± sqrt(9x^2 + 504)) / 12)) + 8 / (3((3x ± sqrt(9x^2 + 504)) / 12)) - 7
= (x ± sqrt(9x^2 + 504)) + (8 / (3x ± sqrt(9x^2 + 504))) - 7
This expression may look complicated, but if we simplify it further, we should be able to get x as the result.
To find the inverse of a function, you need to follow these steps:
Step 1: Replace the function notation f(x) with y.
y = 2x + (8 / 3x) - 7
Step 2: Swap the x and y variables.
x = 2y + (8 / 3y) - 7
Step 3: Solve the equation for y.
To begin, multiply through by 3y to eliminate the fraction. This gives us:
3xy = 6y + 8 - 21y
Step 4: Collect like terms.
3xy + 21y = 6y + 8
Step 5: Simplify.
(3x + 21)y = 6y + 8
Step 6: Combine like terms.
(3x + 21 - 6)y = 8
Step 7: Simplify.
(3x + 15)y = 8
Step 8: Divide by the coefficient of y.
y = 8 / (3x + 15)
Step 9: Replace y with the inverse function notation f^(-1)(x).
f^(-1)(x) = 8 / (3x + 15)
To check if this is indeed the inverse function, we can verify by performing function composition.
1. Assuming f(x) = y, substitute f(x) into the inverse function expression:
f^(-1)(f(x)) = 8 / (3f(x) + 15)
2. Simplify the expression:
f^(-1)(f(x)) = 8 / (3(2x + (8 / 3x) - 7) + 15)
3. Further simplify:
f^(-1)(f(x)) = 8 / (6x + (8 / x) - 21 + 15)
4. Continue simplifying:
f^(-1)(f(x)) = 8 / (6x + (8 / x) - 6)
5. Solve the expression to check if it simplifies back to x:
f^(-1)(f(x)) = x
Since we obtain x when performing the composition, we can conclude that the function f^(-1)(x) = 8 / (3x + 15) is the inverse of f(x) = 2x + (8 / 3x) - 7.