Question

Let f left-parenthesis x right-parenthesis equals 2 x squared plus x minus 3 and g left-parenthesis x right-parenthesis equals x minus 1. Perform the indicated operation, then find the domain.left-parenthesis f times g right-parenthesis left-parenthesis x right-parenthesisA) 2 x cubed minus x squared minus 4 x plus 3; domain: all real numbersB) 2 x cubed plus x squared minus 3 x; domain: all real numbersC) 2 x cubed plus x squared plus 4 x minus 3; domain: negative real numbersD) 2 x squared plus x minus 3 x plus 3; domain: positive real numbers

Answer 1

Given the functions:

`f\mleft(x\mright)=2x^2+x-3`
`g(x)=x-1`

You need to multiply them in order to find:

`(f\cdot g)(x)`

Then, you need to set up:

`(f\cdot g)(x)=(2x^2+x-3)(x-1)`

Now you need to simplify it by applying the Distributive Property:

`(f\cdot g)(x)=(2x^2)(x)+(x)(x)-(3)(x)+(2x^2)(-1)+(x)(-1)-(3)(-1)`
`(f\cdot g)(x)=^{}2x^3+x^2-3x-2x^2-x+3`

Adding the like terms (the terms that have the same variable with the same exponent), you get:

`(f\cdot g)(x)=^{}2x^3-x^2-4x+3`

By definition, the Domain of a function is the set of all the possible input values for which the function is defined.

In this case, you can identify that the function obtained is a Polynomial Function because all its exponents are positive integers. By definition, the domain of all Polynomial Functions is:

`Domain\colon All\text{ }Real\text{ }Numbers`

Hence, the answer is: First option.