Inverse of Piecewise Function

[nycmathguy]

Section 1.9
Question 64

How do I determine if a piecewise function has an inverse? Can you work out 64 in step by step fashion?

 

[MathLover1]

64. Range and Inverse of a Piecewise Function

you have y= -x when x≤0
the inverse function is obtained by switching x and y, which is to say when 0≤y

and y=x^2-3x when x>0,

the inverse
To find the inverse of a function expressed as y= an expression in x
rearrange the expression into the form: x= an expression in y

y=x^2-3x
x=y^2-3y
x=(y^2-3y +b^2)-b^2
x=(y^2-3y +(3/2)^2)-(3/2)^2
x=(y-3/2)^2-9/4 where y>=0

or, y= an expression in x
x+9/4=(y-3/2)^2
sqrt(x+9/4)=y-3/2
y=3/2+sqrt(x+9/4) or y=3/2-sqrt(x+9/4)
f^-1(x)=3/2+sqrt(x+9/4) or f^-1(x)=3/2-sqrt(x+9/4) where x>=-9/4

 

[MathLover1]

64.
Range and Inverse of a Piecewise Function

you have y= -x when x≤0
the inverse function is obtained by switching x and y, which is to say when 0≤y

and y=x^2-3x when x>0,

the inverse
To find the inverse of a function expressed as y= an expression in x
rearrange the expression into the form: x= an expression in y

y=x^2-3x
x=y^2-3y
x=(y^2-3y +b^2)-b^2
x=(y^2-3y +(3/2)^2)-(3/2)^2
x=(y-3/2)^2-9/4 where y>=0

or, y= an expression in x
x+9/4=(y-3/2)^2
sqrt(x+9/4)=y-3/2
y=3/2+sqrt(x+9/4) or y=3/2-sqrt(x+9/4)
f^-1(x)=3/2+sqrt(x+9/4) or f^-1(x)=3/2-sqrt(x+9/4) where x>=-9/4

 

[nycmathguy]

64.
Range and Inverse of a Piecewise Function

you have y= -x when x≤0
the inverse function is obtained by switching x and y, which is to say when 0≤y

and y=x^2-3x when x>0,

the inverse
To find the inverse of a function expressed as y= an expression in x
rearrange the expression into the form: x= an expression in y

y=x^2-3x
x=y^2-3y
x=(y^2-3y +b^2)-b^2
x=(y^2-3y +(3/2)^2)-(3/2)^2
x=(y-3/2)^2-9/4 where y>=0

or, y= an expression in x
x+9/4=(y-3/2)^2
sqrt(x+9/4)=y-3/2
y=3/2+sqrt(x+9/4) or y=3/2-sqrt(x+9/4)
f^-1(x)=3/2+sqrt(x+9/4) or f^-1(x)=3/2-sqrt(x+9/4) where x>=-9/4




View attachment 272

Beautiful done. You are amazing. I'm very happy that we met online.

 

[MathLover1]

 

[nycmathguy]

I will keep the notes and work on a few inverse of piecewise functions when time allows. I don't feel good. There's something going on in my body. I hope it's not prostate cancer.