How to find g(x+1) given g(x)=x3−2x2?

Question

inventivx64 · Accepted Answer

A function returns an output after accepting an input.
In this function

g (x)

, you input

x

and it returns

x^{3} - 2 x^{2}

. We can also look at this as this: you input

x

(what's enclosed by the parentheses) and this function

g (x)

takes the input (x), cubes it (

x^{3}

), subtracts two times the input squared (

- 2 x^{2}

). Putting this all together, this particular function

g (x)

takes in an input

x

and returns the expression

x^{3} - 2 x^{2}

, with

x

representing the input (which could be anything).
So what exactly does the question mean by

g (x + 1)

? What this means is that the entire expression

x + 1

is the input! So, because

g (x) = x^{3} - 2 x^{2}

so,

g (x + 1) = {(x + 1)}^{3} - 2 {(x + 1)}^{2}

So now all we have to do is simplify this expression using algebra!
We have

g (x + 1) = {(x + 1)}^{3} - 2 {(x + 1)}^{2}

g (x + 1) = (x + 1) (x + 1) (x + 1) - 2 (x^{2} + 2 x + 1)

g (x + 1) = (x + 1) (x^{2} + 2 x + 1) - 2 (x^{2} + 2 x + 1)

Here, we see that we have a

(x^{2} + 2 x + 1)

term in common on both sides of the minus sign. We invoke the distributive property:

g (x + 1) = (x^{2} + 2 x + 1) (x + 1 - 2)

g (x + 1) = (x^{2} + 2 x + 1) (x - 1)

And we're done!
Alternatively, you could also write the answer like this:

g (x + 1) = {(x + 1)}^{2} (x - 1) = (x^{2} - 1) (x + 1)

How to find g(x+1) given g(x)=x^3-2x^2?