Which would be the inverse of this:
Let f(x)=(x+3)(2x-5)
f^ (-1) x = (5x - 3)/ (2x + 1)
f^ (-1) x = (5x + 3)/ (2x - 1)
Could you check if there wasn't a typo, namely,
"Let f(x)=(x+3)/(2x-5)... "
If that's the case, it's the second response.
To double check,
let g(x)=f-1(x)=(5x+3)/(2x-1)
Evaluate
f(g(x))
= ((5x+3)/(2x-1)+3)/((2(5x+3))/(2x-1)-5)
= x
Let y = f(x)
y = (x + 3)/ (2x - 5)
Switch x and y.
x = (y + 3)/ (2y - 5)
Multiply both sides by (2y - 5).
2xy - 5x = y + 3
Subtract 3 from both sides.
2xy - 5x - 3 = y
Subtract 2xy from both sides.
-5x - 3 = -2xy + y
Factor the right side.
-5x - 3 = y (-2x + 1)
Divide both sides by (-2x + 1).
(-5x - 3)/ (-2x + 1) = y
(5x - 3)/ (2x + 1) = y
Now we replace y with the inverse function notation: f^ (-1) x.
f^ (-1) x = (5x + 3) / (2x - 1)
This my work to reflect the answer. Is it correct? Thanks!
Yes, the calculation is correct.
Note: You may not have noticed that you omitted the division sign in the initial post.
Thanks for pointing that out!
You're welcome!
To find the inverse function of f(x) = (x+3)(2x-5), we need to switch the roles of x and y in the equation and solve for y.
Step 1: Write the original equation:
f(x) = (x+3)(2x-5)
Step 2: Replace f(x) with y:
y = (x+3)(2x-5)
Step 3: Swap x and y:
x = (y+3)(2y-5)
Step 4: Solve for y:
Expand the equation:
x = 2y^2 - 5y + 6y - 15
x = 2y^2 + y - 15
Rearrange the equation:
2y^2 + y - 15 - x = 0
Step 5: Apply the quadratic formula:
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solution for x is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Applying this formula to our equation, where a = 2, b = 1, and c = -15, we can solve for y:
y = (-1 ± √(1^2 - 4(2)(-15))) / (2(2))
y = (-1 ± √(1 + 120)) / 4
y = (-1 ± √121) / 4
Simplifying further:
y = (-1 ± 11) / 4
This gives two possible values for y:
y = (-1 + 11) / 4 = 10/4 = 2.5
y = (-1 - 11) / 4 = -12/4 = -3
Step 6: Write the inverse function:
Now that we have the values of y, we can write the inverse function.
f^(-1)(x) = 2.5 or f^(-1)(x) = -3
Therefore, the inverse of f(x) = (x+3)(2x-5) is f^(-1)(x) = 2.5 or f^(-1)(x) = -3.