(Solved): 1. The superposition principle applies when you have two functions, \( f(x) \) and \( g(x) \), and ...

1. The superposition principle applies when you have two functions, \( f(x) \) and \( g(x) \), and you are trying to obtain the graph of a. \( y=f(x)+g(x) \) b. \( y=\frac{f(x)}{g(x)} \) c. \( y=f(x) g(x) \) d. \( y=f(g(x)) \) 2. Given the functions \( f(x)=x^{2}+1 \) and \( g(x)=3-x \), determine an equation for the combined function \( y=f(x)+g(x) \). a. \( y=x^{2}-x+2 \) b. \( y=x^{2}-x+4 \) c. \( y=x^{2}+x+4 \) d. \( y=x^{2}+x-2 \) 3. Given the functions \( f(x)=x^{2}+1 \) and \( g(x)=3-x \), determine an equation for the combined function \( y=f(x) g(x) \). a. \( y=-x^{3}+3 x^{2}-x+3 \) b. \( y=-x^{3}+3 x^{2}+x-3 \) c. \( y=x^{3}+3 x^{2}-x+3 \) d. \( y=-x^{3}+2 x^{2}-x+3 \) 4. Given the functions \( f(x)=x^{3}-x \) and \( g(x)=x-1 \), determine an cquation for the combined function \( y=\frac{f(x)}{g(x)} \). a. \( y=x^{2}+x, x \neq 1 \) b. \( y=\frac{x}{x-1}, x \neq 1 \) c. \( y=x^{2}-x, x \neq 1 \) d. \( y=\frac{x^{2}-x}{x-1}, x=1 \) 5. Given the functions \( f(x)=x^{2}-x \) and \( g(x)=x-1 \), determine an equation for the composite function \( y=f(g(x)) \). a. \( y=x^{2}-3 x+1 \) b. \( y=x^{2}-3 x+2 \) c. \( y=x^{2}-x-1 \) d. \( y=x^{2}-x+1 \) 6. Given the functions \( f(x)=\sin x \) and \( g(x)=x \), determine the domain of the combined function \( y=f(x)+g(x) \). a. \( \{x \in \mathbb{R},-2 \pi \leq x \leq 2 \pi\} \) b. \( \{x \in \mathbb{R}\} \) c. \( \{x \in \mathbb{R},-1 \leq x \leq 1\} \) d. cannot be determined 7. In general, the zeros of a function \( f(x) \) appear on the graph of \( y=f(x) g(x) \) as \( A x \)-intercepts b.holes c. vertical asymptotes d. local extreme points 8. The zeros of a function \( g(x) \) appear on the graph of \( y=\frac{f(x)}{g(x)} \) as a. \( x \)-intercepts b. holes c. vertical asymptotes d. B or \( \mathrm{C} \)