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Abstract Algebra

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Operations on Functions: Basic Operations on Standard Functions

ABSALG-V2DKR1

Let $f:D_f\rightarrow \mathbb{R}$ and $g: D_g\rightarrow \mathbb{R}$ be two functions with domains $D_f$, $D_g$ in $\mathbb{R}$ respectively.

We define the following basic operations on functions (with their corresponding domains):

Addition $\quad(f+g)(x):=f(x)+g(x)$ (on $D_f\cap D_g$)
Subtraction $\quad(f-g)(x):= f(x)-g(x)$ (on $D_f\cap D_g$)
Multiplication $\quad(f\cdot g)(x)=f(x)g(x)$ (on $D_f\cap D_g$)
Division $\quad(f/g)(x):=f(x)/g(x)$ (on $D_f\cap D_g\cap \{x\in D_g\mid g(x)\not=0\}$)
Composition $\quad (f\circ g)(x):=f(g(x))$ (on $g(D_g)\cap D_f$)

Let $g(x)=\sqrt{x}$, $h(x)=x$, $f(x)=x^2$.

Which of the following equals the function with domain $\{r\in\mathbb{R}\mid r > 0\}$ given by:

$$g\circ\left(h\cdot\left(f\circ\left(g-\cfrac1g\right)\right)\right)$$

A

$\sqrt{x^{7/2}-x^{5/2}}$

B

$x-1$

C

$\sqrt{x-2+x^{-1}}$

D

$\sqrt{x(x-1)^2}$