How do I find the domain and range of y= -(x-2)^2+3?​

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined. The range is the set of all valid y values.

Step-by-step explanation:

D=(-\infty,3)\cup(3,\infty) [x|x\neq3]D=(−∞,3)∪(3,∞)[x∣x



=3]

R=(-\infty,-1)\cup(-1,\infty) [y|y\neq-1]R=(−∞,−1)∪(−1,∞)[y∣y



=−1]

Given : f(x)=\frac{x-2}{3-x}f(x)=

3−x

x−2

To find : The domain and range of the real function

Solution :

To find domain : Equate the denominator to zero

f(x)=\frac{x-2}{3-x}f(x)=

3−x

x−2

Denominator (3-x)=0

x=3

This means at x=3 function is not defined

And by definition of domain - The domain is where the function is not defined.

Domain is D=(-\infty,3)\cup(3,\infty) [x|x\neq3]D=(−∞,3)∪(3,∞)[x∣x



=3]

Range

Put f(x)=y

y=\frac{x-2}{3-x}y=

3−x

x−2

x=\frac{3y-2}{y+1}x=

y+1

3y−2

Range is the set of value that correspond to domain

Equate the denominator to zero

x=\frac{3y-2}{y+1}x=

y+1

3y−2

Denominator (y+1)=0

y=-1

This means at y=-1 function is not defined

Range is R=(-\infty,-1)\cup(-1,\infty) [y|y\neq-1]R=(−∞,−1)∪(−1,∞)[y∣y



=−1]

step-by-step explanation: