Question 222993
In all these equations, let y = f(x).


To find the inverse function, solve for x and then replace y with x and x with y.


alternatively you can replace y with x and x with y first, and then solve for y.


Either way you'll get the same answer.


selection A

y = x+3
Solve for x to get:
x = y-3
Replace x with y and y with x to get:
y = x-3 which is the inverse function of y = x+3


Selection B
y = x/2
Solve for x to get:
x = 2y
Replace x with y and y with x to get:
y = 2x which is the inverse function of y = x/2


Selection C
y = -x-2
Solve for x to get:
x = -y-2
Replace x with y and y with x to get:
y = -x-2 which is the inverse function of y = -x-2


Selection D
y= sqrt(x+2)
Solve for x as follows:
Square both sides to get:
y^2 = x+2
Subtract 2 from both sides to get:
y^2-2 = x which is the same as:
x = y^2-2
Replace x with y and y with x to get:
y = x^2-2 which is the inverse function of y = sqrt(x+2)


Selection C inverse equation is the same as the original equation.



graph of this equation is shown below:


{{{graph(300,300,-5,5,-5,5,-x-2,x)}}}


The line y = -x-2 is a reflection about the line y = x which is a definition of inverse function.


Take any point (x,y) on the line y = -x-2 above the line y = x.   The opposite point (y,x) on the line y = -x-2 below the line will be the same distance from the line y = x.


example:


let x = -2
then y = -(-2)-2 = 2-2 = 0
your coordinate point is (-2,0).


now let x = 0
then y = 0-2 = -2
your coordinate point is (0,-2)


the point (-2,0) is a reflection of the point (0,-2) about the line y = x


to prove that the distance between these points and the line y = x is the same, we need to find the point of intersection between these two lines.


the point of intersection with the line y = x would be (-1,-1) as shown on the graph.


The distance between the point (-2,0) and (-1,-1) is given by the equation:


{{{sqrt((-1-(-2))^2 + (-1-0)^2)}}} which equals {{{sqrt(1^2+(-1)^2) = sqrt(2)}}}


The distance between the point (0,-2) and (-1,-1) is given by the equation:


{{{sqrt((-1-0)^2 + (-1-(-2))^2)}}} which equals {{{sqrt((-1)^2+1^2) = sqrt(2)}}}


The point (-2,0) is a reflection of the point (0,-2) about the line y = x which is a requirement of inverse functions.


Selection C is your answer.