the domain is all values of x except x = 4/7.
the graph looks like this:
there is a vertical asymptote at x = 4/7.
all values of x in the domain map to a single value of y.
there are no multiple values of x for the same value of y.
the graph is therefore 1 to 1 for all values of x in its domain.
the function can be shown as, and is graphed as:
y = (-4x+8)/(7x-4)
the inverse function would be:
x = (-4y+8)/(7y-4)
if you had graphing software that would graph that function the way it's shown, you would see that it is the inverse function.
alternately, you can replace x with y and y with x in that function and then solve for y.
i haven't always been successful in doing that, but i was successful with this function.
start with the inverse function of x = (-4y+8)/(7y-4)
you are looking to solve for y in this equation.
multiply both sides of this equation by (7y-4) to get:
x * (7y-4) = -4y + 8
simplify by distributing the multiplication to get:
x * 7y - 4x = -4y + 8
add 4y to both sides of this equation and add 4x to both sides of this equation to get:
x * 7y + 4y = 4x + 8
factor out the y on the left side of the equation to get:
y * (7x + 4) = 4x + 8
divide both sides of the equation by (7x + 4) to get:
y = (4x+8)/7x+4)
that's your inverse equation after solving it for y.
the graph of your original equation plus the inverse equation is shown below:
the original function is in red and the inverse function is in blue.
the vertical asymptote of the original function is at x = 4/7.
the horizontal asymptote of the inverse function is at y = 4/7
to show that they are inverse functions, i graphed the line y = x and then graphed lines vertical to the line y = x at y = -x + 8 and y = -x-4.
these lines vertical to the line y = x intersect with the original grpah and its inverse and show that (x,y) in the original graph has its counterpart at (y,x) in the inverse graph.
that is used to confirm that the graphs are inverse with each other.
one example is that the point (.675,7.325) on the red grpah has itse counterpart at (7.325,0.675) on the vlue graph.
bottom line is that the function is 1 to 1 and it's inverse has been determined and mapped as shown.