– GIVEN F OF X

=THE QUANTITY X + 2 SQUARED, WE WANT TO DETERMINE THE DOMAIN

SO F OF X IS INCREASING AND 1 TO 1. WE ALSO WANT TO GIVE THE RANGE

AND USE INTERVAL NOTATION. SO HERE’S A GRAPH

OF OUR FUNCTION F OF X. NOTICE IF WE DON’T RESTRICT

THE DOMAIN, THIS FUNCTION IS NOT ONE TO ONE BECAUSE HORIZONTAL LINES

WOULD INTERSECT THIS GRAPH IN MORE THAN ONE POINT. NOTICE THE FUNCTION

IS ALSO DECREASING ON THE LEFT AND INCREASING ON THE RIGHT. SO NOTICE IF WE CONSIDER

THIS FUNCTION ONLY FROM THE VERTEX

TO THE RIGHT, THE FUNCTION IS INCREASING AND IT’S ALSO 1 TO 1

BECAUSE HORIZONTAL LINES WOULD ONLY INTERSECT THIS HALF

OF THE GRAPH AT ONE POINT. SO NOW WE’LL DETERMINE

THE DOMAIN AND RANGE IF WE ONLY WANT

THIS HALF OF THE GRAPH. WELL, THE DOMAIN IS A SET

OF ALL POSSIBLE X VALUES, SO IF WE PROJECT THIS GRAPH

UNDER THE X AXIS, NOTICE HOW THE DOMAIN WOULD BE

FROM -2 TO THE RIGHT OR FROM -2 TO INFINITY. AND IT WOULD INCLUDE THE VERTEX, SO WE’LL INCLUDE -2

IN THE DOMAIN. SO THE DOMAIN,

USING INTERVAL NOTATION, WOULD BE FROM -2 TO INFINITY. AND IT’S CLOSED ON -2

MEANING IT INCLUDES -2. WE COULD ALSO EXPRESS THIS

USING INEQUALITIES AS X IS GREATER THAN OR=TO -2. NOW LET’S CONSIDER THE RANGE. THE RANGE IS A SET

OF ALL POSSIBLE Y VALUES OR OUTPUTS OF THIS FUNCTION

ON THE RESTRICTED DOMAIN. WELL, IF WE PROJECT THIS GRAPH

ON TO THE Y AXIS, NOTICE HOW THE SMALLEST Y VALUE

WOULD BE 0, AND FROM THERE IT INCREASES

UPWARD TOWARD POSITIVE INFINITY. NOW THE RANGE WOULD BE THE

INTERVAL FROM 0 TO INFINITY, CLOSED ON 0

MEANING IT INCLUDES 0, OR WE COULD SAY Y

IS GREATER THAN OR=TO 0. WITH THIS RESTRICTION, THE

FUNCTION F IS NOW ONE TO ONE, SO WE CAN FIND F INVERSE OF X. TO DO THIS, LET’S FIRST WRITE

THE ORIGINAL FUNCTION REPLACING F OF X WITH Y, SO WE’D HAVE Y=

THE QUANTITY X + 2 SQUARED. AND THEN TO FIND THE INVERSE, WE INTERCHANGE

THE X AND Y VARIABLES, AND THEN SOLVE FOR Y. SO WE HAVE X=

QUANTITY Y + 2 SQUARED. AND NOW WE’LL SOLVE THIS FOR Y. THE FIRST STEP WE’LL TAKE

THE SQUARE ROOT OF BOTH SIDES OF THIS EQUATION. SO WE’D HAVE THE SQUARE ROOT

OF X EQUALS THE SQUARE ROOT OF THE QUANTITY

Y + 2 SQUARED. SO WE HAVE THE SQUARE ROOT OF X

EQUALS– NORMALLY THIS WOULD BE

THE OPPOSITE VALUE OF Y + 2, BUT BECAUSE OF THE RESTRICTIONS

HERE WE DON’T HAVE TO WORRY

ABOUT THAT. THIS WOULD JUST BE Y + 2. LAST STEP WE’LL SUBTRACT 2

ON BOTH SIDES, WE HAVE THE SQUARE ROOT OF X – 2

=Y. THIS IS OUR INVERSE FUNCTION

SOLVE FOR Y. SO WE’LL GO AHEAD AND REPLACE

Y WITH F INVERSE OF X. F INVERSE OF X IS EQUAL

TO THE SQUARE ROOT OF X – 2. WE’RE ALSO ASKED

TO GIVE THE DOMAIN AND RANGE. LET’S GO AHEAD AND DO THAT. BECAUSE THIS IS THE INVERSE

OF FUNCTION F, THE DOMAIN OF F IS GOING TO BE

THE RANGE OF F INVERSE, AND THE RANGE OF F WILL BE

THE DOMAIN OF F INVERSE. SO THE DOMAIN WILL BE

FROM 0 TO INFINITY, CLOSED ON 0, AND THE RANGE WILL BE THE

INTERVAL FROM -2 TO INFINITY. LET’S GO AHEAD AND FINISH

BY VERIFYING THIS GRAPHICALLY. WE KNOW THAT

IF WE GRAPH FUNCTION F ON THE RESTRICTED DOMAIN AND WE GRAPH THE INVERSE

FUNCTION ON ITS DOMAIN. THE TWO FUNCTIONS SHOULD BE

SYMMETRICAL ACROSS THE LINE Y=X. AND HERE’S THE GRAPH

OF THE ORIGINAL FUNCTION ON THE RESTRICTED DOMAIN, AND HERE’S THE GRAPH

OF THE INVERSE FUNCTION GRAPHED OVER ITS DOMAIN. NOW WE CAN SEE THAT

THESE TWO GRAPHS ARE SYMMETRICAL ACROSS THE LINE Y=X. OKAY, I HOPE YOU FOUND THIS

HELPFUL.

## 13 Comments

## mishellee

Thank you so much! ðŸ˜€

## 95duckie

Really helpful, thanks so much!

## alfred vish

best explanation ever…

## Marangeli Torres

Hi, why not the plus minus square root of x when we are solving for y video at 2:25?

## Marangeli Torres

Also, what hardware (tablet), software do you use to produce these videos?Â I want to make some

## Tony M

This really helped ALOT thank you

## mmmay

Why is it that I learn so much more from YouTube videos like this, than from my actual pre-calc class?!ðŸ˜‘ðŸ˜‚ Thank you so much for the video and explanation!!!

## C. James McMurray

I did find this helpful!

## No Name

Explained better in 4 min than my teacher in 50 min

## Glenda Cuellar

I have only one question…Why did you chose the right side of the original graph rather than the left one?

## Aditi Bhat

thank you!! this is SO helpful! = )

## Denni

yes, it is helpful.

## Cole Johnson

do not understand how to find which intervals can make a graph 1 to 1