Question

Let f(x) = x2, g(x) = √x + 3.

a. Show that (f(x + 3)) = |x + 3| + 3.
b. Does (x) = |x + 3| + 3 = (x) = |x| + 6? Graph them on the same coordinate plane

Answer 1

Answer with Step-by-step explanation:

We are given that f(x)=
`x^2`

`g(x)=√(x)+3`

a.We have to show that
`g(f(x+3))=\mid{x+3}\mid +3`

`g(f(x+3))=√((x+3)^2)+3`

When we remove square root then we take plus minus therefore we write in modulus.

Therefore,
`g(f(x+3))=\mid {x+3}\mid+3`

Hence, proved.

b.We have to find that

`g(x)=\mid{x+3}\mid+3=\mid x\mid+6`

Substitute x=-3

Then, we get

`g(x)=\mid{-3+3}\mid+3=3`

`g(x)=\mid{-3}\mid+6=9`

`3\\eq 9`

`\mid{x+3}\mid +3=\mid x\mid+6` for
`x\geq 0`

But,
`\mid{x+3}\mid+3\\eq \mid x\mid+6` for x<0

Hence,
`\mid{x+3}\mid+3\\eq \mid{x}+6` for all x.