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)=$x^2$

$g(x)=\sqrt{x}+3$

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

$g(f(x+3))=\sqrt{(x+3)^2}+3$

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

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

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

$3\neq 9$

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

But, $\mid{x+3}\mid+3\neq \mid x\mid+6$ for x<0

Hence, $\mid{x+3}\mid+3\neq \mid{x}+6$ for all x.