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For the following problems, consider the functions
• $f:\mathbb {N}\rightarrow \mathbb {R}$ defined by $f(x) = x + \dfrac {1}{x}$
• $g:\mathbb {N}\rightarrow \mathbb {R}$ defined by $g(x) = 2x$
• $h:\mathbb {R}\rightarrow \mathbb {R}$ defined by $h(x) = 2x$
1 : What is $(f\circ g)(x)$?
$(f\circ g)(x) = 2x + \dfrac {1}{2x}$ $(f\circ g)(x) = 2x + \dfrac {2}{x}$ $(f\circ g)(x)$ is undefined $(f\circ g)(x) = 4x$
2 : What is $(g\circ f)(x)$?
$(g\circ f)(x) = 2x + \dfrac {1}{2x}$ $(g\circ f)(x) = 2x + \dfrac {2}{x}$ $(g\circ f)(x)$ is undefined $(g\circ f)(x) = 4x$
3 : What is $(f\circ h)(x)$?
$(f\circ h)(x) = 2x + \dfrac {1}{2x}$ $(f\circ h)(x) = 2x + \dfrac {2}{x}$ $(f\circ h)(x)$ is undefined $(f\circ h)(x) = 4x$
4 : What is $(h\circ f)(x)$?
$(h\circ f)(x) = 2x + \dfrac {1}{2x}$ $(h\circ f)(x) = 2x + \dfrac {2}{x}$ $(h\circ f)(x)$ is undefined $(h\circ f)(x) = 4x$

For the next set of problems consider the following functions:

• $f:\mathbb {R}\rightarrow \mathbb {R}$ defined by $f(x) = x^2$
• $g:\mathbb {R}^+\rightarrow \mathbb {R}^+$ defined by $g(x) = \sqrt {x}$
• $h:\mathbb {R}\rightarrow \mathbb {R}$ defined by $h(x) = \sqrt [3]{x}$
5 : What is $(f\circ g)(x)$?
$(f\circ g)(x) = x$ $(f\circ g)(x) = x^2$ $(f\circ g)(x)$ is undefined $(f\circ g)(x) = \sqrt {x}$
6 : What are the domain and codomain of $(f\circ g)(x)$?
Domain: $\mathbb {R}$ and Codomain: $\mathbb {R}^+$. Domain: $\mathbb {R}^+$ and Codomain: $\mathbb {R}$. Domain: $\mathbb {R}$ and Codomain: $\mathbb {R}$. Domain: $\mathbb {R}+$ and Codomain: $\mathbb {N}$.
7 : What is $(g\circ f)(x)$?
$(g\circ f)(x) = x$ $(g\circ f)(x) = |x|$ $(g\circ f)(x)$ is undefined $(g\circ f)(x) = x^2$
8 : What are the domain and codomain of $(g\circ f)(x)$?
Domain: $\mathbb {R}$ and Codomain: $\mathbb {R}^+$. Domain: $\mathbb {R}^+$ and Codomain: $\mathbb {R}$. Domain: $\mathbb {R}$ and Codomain: $\mathbb {R}$. Domain: $\mathbb {R}+$ and Codomain: $\mathbb {N}$.
9 : What is $(f\circ h)(x)$?
$(f\circ h)(x) = \left (\sqrt [3]{x}\right )^2$ $(f\circ h)(x) = x$ $(f\circ h)(x)$ is undefined $(f\circ h)(x) = |x|$
10 : What are the domain and codomain of $(f\circ h)(x)$?
Domain: $\mathbb {R}$ and Codomain: $\mathbb {R}^+$. Domain: $\mathbb {R}^+$ and Codomain: $\mathbb {R}$. Domain: $\mathbb {R}$ and Codomain: $\mathbb {R}$. Domain: $\mathbb {R}+$ and Codomain: $\mathbb {N}$.
11 : What is $(h\circ f)(x)$?
$(h\circ f)(x) = x$ $(h\circ f)(x)$ is undefined $(h\circ f)(x) = |x|$ $(h\circ f)(x) = \sqrt [3]{x^2}$
12 : What are the domain and codomain of $(h\circ f)(x)$?
Domain: $\mathbb {R}$ and Codomain: $\mathbb {R}^+$. Domain: $\mathbb {R}^+$ and Codomain: $\mathbb {R}$. Domain: $\mathbb {R}$ and Codomain: $\mathbb {R}$. Domain: $\mathbb {R}+$ and Codomain: $\mathbb {N}$.