If #r(x) = 2-x^2# and #w(x) = x-2#, what is the range of #(W*r)(x)#?

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Answer 1 · 2018-06-25T20:50:21

#color(blue)((-oo,oo)#

#(w*r)(x)=>w(x)*r(x)#

#w(x)=x-2#

#r(x)=2-x^2#

#w(x)*r(x)=(x-2)(2-x^2)=-x^3+2x^2+2x-4#

This is a polynomial so it has domain #RR#

To find the range, we observe what happens as #x->+-oo#.

For polynomials we only need to look at the leading coefficient and the degree of the polynomial. In this case:

#-x^3#

as #x->-oo# ,#color(white)(8888)-x^3->oo#

as #x->oo# ,#color(white)(8888)-x^3->-oo#

The range of #(w*r)(x)# is therefore:

#(-oo,oo)#