Find the domain and range of the following functions interval notation #f(x)=x^2+4x# and #g(x)=6/(1-x)# ?

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Answer 1 · 2017-02-02T11:43:45

#(1):f: RR to [-4,oo).#

#(2):g: (-oo,1)uu(1,oo) to (-oo,0)uu(0,oo).#

We will discuss the Soln. in #RR.#

#f(x)=x^2+4x.#

So, to operate #f#, we can take choose any #x# from #RR#,

meaning that, the Domain is #RR#.

Next, we note that, #f(x)=x^2+4x=(x^2+4x+4)-4=(x+2)^2-4.#

Also, #AA x in RR, (x+2)^2>=0;" adding "-4, (x+2)^2-4>=-4.#

#rArr f(x)>=-4#

#:." The Range is "[-4, oo).#

Regarding, #g(x)=6/(1-x)," we can not divide by 0; so, "(1-x)ne0.#

In other words, the Domain of

#g" is "RR-{1}=(-oo,1)uu(1,oo)#.

Further, since, #g(x)=6(1/(1-x)), and, AA x in RR-{1}, 1/(1-x)ne0#

#rArr g(x)ne0.#

#rArr " the Range of g is "RR-{0}=(-oo,0)uu(0,oo).#

Enjoy Maths.!