The inverse of f(x)=2x-1/x+5 may be writeen in the form f^-1(x)=ax+b/cx+d where a,b,c, and d are real numbers. Find a/c. Thank you!
You probably meant:
f(x) = (2x-1)/(x+5)
I will assume that
then
y = (2x-1)/(x+5)
we form the inverse by interchanging x and y
x = (2y-1)/(y+5)
xy + 5x = 2y -1
now solve this for y
xy - 2y = -5x - 1
y(x-2) = -5x - 1
y = f^-1 (x) = (-5x - 1)/(x - 2)
so by comparison,
a = -5
b = -1
c = 1
d = -2
of course we could also write our inverse as
f^-1 (x) = (5x+1)/(-x+2)
all you would have to do is take the opposite of all the a,b, ...'s
To find the inverse of the function f(x) = (2x - 1)/(x + 5), we can follow these steps:
Step 1: Replace f(x) with y:
y = (2x - 1)/(x + 5)
Step 2: Swap x and y:
x = (2y - 1)/(y + 5)
Step 3: Solve for y:
Cross multiply to get rid of the denominators:
x(y + 5) = 2y - 1
xy + 5x = 2y - 1
xy - 2y = -5x - 1
y(x - 2) = -5x - 1
y = (-5x - 1)/(x - 2)
Therefore, the inverse function f^-1(x) can be written in the form: f^-1(x) = (-5x - 1)/(x - 2).
Comparing the form f^-1(x) = ax + b/cx + d with f^-1(x) = (-5x - 1)/(x - 2), we can determine that:
a = -5
c = 1
Therefore, a/c = -5/1 = -5.
So, a/c = -5.
To find the inverse of the function f(x) = (2x - 1) / (x + 5) and express it in the form f^-1(x) = (ax + b) / (cx + d), we can follow these steps:
Step 1: Replace f(x) with y.
y = (2x - 1) / (x + 5)
Step 2: Swap x and y.
x = (2y - 1) / (y + 5)
Step 3: Solve the equation for y.
Cross-multiplying:
x(y + 5) = 2y - 1
xy + 5x = 2y - 1
Moving all terms involving y to one side:
xy - 2y = -5x - 1
Factoring out y:
y(x - 2) = -5x - 1
Dividing both sides by (x - 2):
y = (-5x - 1) / (x - 2)
Step 4: Replace y with f^-1(x) to express the inverse function in the form f^-1(x) = (ax + b) / (cx + d).
f^-1(x) = (-5x - 1) / (x - 2)
Comparing this with the desired form f^-1(x) = (ax + b) / (cx + d), we can see that a = -5, b = -1, c = 1, and d = -2.
Therefore, a/c = -5/1 = -5.
Hence, a/c = -5.