# Thread: Transformation - Inverse Question

1. ## Transformation - Inverse Question

For the following functions

Equation --> y=x^2-3

a).Find Inverse f^-1
b).graph f(x) and its inverse

c).restric the domain of f so that f^-1 is also a function

d). with the domain of f restricted, sketch a graph of f and f^-1

k so what i did is first i did the inverse:

f(x)=x^2+3
so what i did is let

x=y^2+3
x-3=y^2
y=+-sqrt/x-3

Then i graphed the normal equation and it turned our fine, but i need help drawing the mirror image(with table of values and steps on how to do so)

also someone help with part c and d in detial please

Thank You

2. Originally Posted by usm_67
For the following functions

Equation --> y=x^2-3

a).Find Inverse f^-1
b).graph f(x) and its inverse

c).restric the domain of f so that f^-1 is also a function

d). with the domain of f restricted, sketch a graph of f and f^-1

k so what i did is first i did the inverse:

f(x)=x^2+3
so what i did is let

x=y^2+3
x-3=y^2
y=+-sqrt/x-3

Then i graphed the normal equation and it turned our fine, but i need help drawing the mirror image(with table of values and steps on how to do so)

also someone help with part c and d in detial please

Thank You
Hello,

1. you somehow changed the term of the function. I'll take the first version for further considerations.

If D is the domain of f and R is the range of f you got:

$f: y = x^2-3,~~D_f=\mathbb{R},~R_f=\{y|y\geq -3\}$

To calculate the inverse of f you change x and y and you have to change domain and range too:

$f^{-1}: x = y^2-3,~~D_{f^{-1}}=\{x|x\geq -3\},~R_{f^{-1}}=\mathbb{R}$ . As you can see the inverse of f is not a function.

to c) Roughly speaking a continuously increasing (or decreasing) function has an inverse function too. Thus you have to divide f into two branches(?):
The decreasing branch: $f: y = x^2-3,~~D_f=\{x|x < 0\},~R_f=\{y|y\geq -3\}$
and the increasing branch: $f: y = x^2-3,~~D_f=\{x|x \geq 0\},~R_f=\{y|y\geq -3\}$

Solve the equation of the inverse of for y and you get the functions and their inverses as:

$f_1: y = x^2-3,~~D_f=\{x|x < 0\},~R_f=\{y|y\geq -3\}~~ \Longrightarrow f_1^{-1}: y=-\sqrt{x+3},~$ $D_{f_1^{-1}}=\{x|x >-3\},~R_{f_1^{-1}}=\{y|y <0\}$

and

$f_2: y = x^2-3,~~D_{f_2}=\{x|x \geq 0\},~R_{f_2}=\{y|y\geq -3\}~~ \Longrightarrow f_2^{-1}: y=\sqrt{x+3},~$ $D_{f_2^{-1}}=\{x|x >-3\},~R_{f_1^{-1}}=\{y|y \geq 0\}$

I've attached a drawing. Corresponding parts of the funcrtion and it's inverse are marked in similar colours.