Question

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

Answer 1

Answer:

Range of (w·r)(x) is all real numbers.

Step-by-step explanation:

The given function are

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

$w(x)=x-2$

We need to find the range of (w·r)(x).

Product of two function is defined as

$(w\cdot r)(x)=w(x)\cdot r(x)$

Substitute the values of given functions.

$(w\cdot r)(x)=(x-2)\cdot (2-x^2)$

$(w\cdot r)(x)=x\cdot (2-x^2)-2\cdot (2-x^2)$

$(w\cdot r)(x)=2x-x^3-4+2x^2$

$(w\cdot r)(x)=-x^3+2x^2+2x-4$

(w·r)(x) is a three degree polynomial. Domain and range of a odd degree polynomial is all real numbers.

Therefore, the range of (w·r)(x) is all real numbers.

Answer 2

First, find the product (w*r)(x):  (w*r)(x) = (x-2)*[2-x^2] = 2x - x^3 - 4 + 2x^2

This is a cubing function.  Since the sign of the cube-of-x term is negative, the graph will begin in Quadrant II and pass through Quadrant IV.  There are no limits on y.  Thus, the range is (-infinity, +infinity).