Question 227824
g(x) = 2x - 5 is the equation of a line and is a function because there is only one value of y for each value of x.


The inverse function of g(x) is found by:


Solving for x.
Replacing x with y and y with x.


Let y = g(x)


Your equation becomes y = 2x - 5


Solve for y.


subtract y from both sides of the equation to get:
0 = 2x - 5 - y
subtract 2x from both sides of the equation to get:
-2x = -y - 5
divide both sides of the equation by -2 to get:
x = y/2 + 5/2


Replace x with y and y with x to get:


y = x/2 + 5/2


If this is the inverse function, then both equations will be a reflection about the line y = x.


A graph of these equations and the equation of y = x is shown below:


{{{graph (600,600,-10,10,-10,10,x/2 + 5/2, 2x - 5, x)}}}


If these are inverse functions, then:


f(g(x) = g(f(x)


Take f(g(x))


f(x) = 2x-5
g(x) = x/2 + 5/2


f(g(x) = f(x/2+5/2) = 2 * (x/2 + 5/2) - 5 = x + 5 - 5 = x


g(f(x) = g(2x-5) = (2x-5)/2 + 5/2 = x - 5/2 + 5/2 = x


We have f(g(x) = g(f(x) which confirms that these equations are inverse functions of each other.


Definition of a function states that you have a 1 to 1 mapping form x to y.


This happens with both these equations as shown in the graph.