We see that for each value of x, f(x)=(2x-5)/(7x+4) has only one value and each value of f(x) can be obtained by only one value of x.
The function has an inverse.
Let y = f(x)=(2x-5)/(7x+4)
express x in terms of y
=> (7x + 4)y = 2x - 5
=> 7xy + 4y = 2x - 5
=> 7xy - 2x = -5 - 4y
=> x(7y - 2) = -5 - 4y
=> x = (5 + 4y)/(2 - 7y)
interchange x and y
=> y = (5 + 4x)/(2 - 7x)
The inverse function f^(-1)(x) = (5 + 4x)/(2 - 7x)
To check if the function has an inverse, we'll must verify if the function is injective. One method to find out if the function is injective is to do 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)