How do you find the domain and range of f(x) = x / (3x(x-1))f(x)=x3x(x1)?

1 Answer
Oct 14, 2015

Domain: RR\{0,1}

Range: RR\{-1/3,0}

Explanation:

Domain:
To find the domain of this rational function, simply look for the values of x that will make the the denominator 0.

color(white)(XXX)3x(x-1)=0

hArrcolor(white)(X)color(blue)(x=0)orcolor(red)(x=1)

Exclude 0 and 1 from the set of real numbers. The domain is RR\{0,1}

Range:
To find the range, start by isolating x.

color(white)(XX)y=x/[3x(x-1)]

color(white)(XX)y=cancelx/[3cancelx(x-1)]

Note: When we cancel x here, we are adding a restriction that x!=0. The graph of this new equation has a horizontal asymptote y=0. Therefore, 0 is removed from the range.

color(white)(XX)y=1/[3(x-1)],x!=0

color(white)(XX)y=1/[3x-3],x!=0

color(white)(XX)y(3x-3)=1,x!=0

color(white)(XX)3xy-3y=1,x!=0

color(white)(XX)3xy=1+3y,x!=0

color(white)(XX)x=(1+3y)/(3y),x!=0

Now take note that x must not be equal to 0 or 1. We will substitute this into the equation to find out the values that y cannot be.

[1]color(white)(X)x=(1+3y)/(3y)

color(white)([1]X)(0)=(1+3y)/(3y)

color(white)([1]X)0=1+3y

color(white)([1]X)3y=-1

color(white)([1]X)y=-1/3

[2]color(white)(X)x=(1+3y)/(3y)

color(white)([2]X)(1)=(1+3y)/(3y)

color(white)([2]X)cancel(3y)=1+cancel(3y)

color(white)([2]X)0!=1

From [1], we know that -1/3 is excluded from the range. [2] doesn't matter. And looking back at our note, 0 is also excluded from the range. Therefore, the range is RR\{-1/3,0}