Determine whether the function f(x)=(2x-5)/(7x+4) has an inverse and , if so , find the inverse.
To verify if the function has an inverse, we'll have to check if the function is one to one.
There are at least 2 ways to determine if a function is one to one. One of them is to graph the function and to make the horizontal line test.
Doing the test, the horizontal line hits the graph in at most one point. If so, the function is one to one.
We'll determine the inverse function:
y = (2x-5)/(7x+4)
Firt, we'll interchange x and y:
x = (2y-5)/(7y+4)
We'll multiply both sides by (7y+4):
x(7y+4) = 2y - 5
We'll remove the brackets using the distributive law:
7xy + 4x = 2y - 5
We'll keep all terms in y to the left side and we'll move the rest to the right side:
7xy - 2y = -4x - 5
We'll factorize by y to the left side:
y(7x - 2) = -4x - 5
We'll divide by (7x - 2):
y = (-4x - 5)/ (7x - 2)
The inverse function is:
f^-1(x) = (-4x - 5)/ (7x - 2)