What is the domain and range of y=(4+x)/(1-4x)?

2 Answers
May 29, 2017

The domain is RR-{1/4}
The range is RR-{-1/4}

Explanation:

y=(4+x)/(1-4x)

As you cannot divide by 0, =>, 1-4x!=0

So,

x!=1/4

The domain is RR-{1/4}

To find the range, we calculate the inverse function y^-1

We interchange x and y

x=(4+y)/(1-4y)

We express y in terms of x

x(1-4y)=4+y

x-4xy=4+y

y+4xy=x-4

y(1+4x)=x-4

y=(x-4)/(1+4x)

The inverse is y^-1=(x-4)/(1+4x)

The range of y is = to the domain of y^-1

1+4x!=0

The range is RR-{-1/4}

May 29, 2017

x inRR,x!=1/4
y inRR,y!=-1/4

Explanation:

"the domain is defined for all real values of x, except"
"those values which make the denominator zero"

"to find excluded values equate the denominator to zero"
"and solve for x"

"solve " 1-4x=0rArrx=1/4larrcolor(red)"excluded value"

rArr"domain is " x inRR,x!=1/4

"to find any excluded values in the range, change the subject"
"of the function to x"

y(1-4x)=4+x

rArry-4xy=4+x

rArr-4xy-x=4-y

rArrx(-4y-1)=4-y

rArrx=(4-y)/(-4y-1)

"the denominator cannot equal zero"

rArr-4y-1=0rArry=-1/4larrcolor(red)" excluded value"

rArr"range is " y inRR,y!=-1/4