Question

Question: Let $\mathit{S}\mathbf{=}\left\{-1,0,2,4,7\right\}$. Find if $\mathit{f}\mathbf{\left(}\mathit{S}\mathbf{\right)}$

$$\begin{gathered} \mathbf{\text{a)f(x)=1}} \\ \mathbf{\text{b)f(x)=2x+1}} \\ \mathbf{\text{c)}}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\lceil x/5\rceil \\ \mathbf{\text{d)}}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\lceil (x^2+1)/3\rceil \end{gathered}$$

Short answer

Answer:

$$\begin{gathered} a)f(S)=\left\{-1,0,2,4,7\right\} \\ b)f\left(S\right)=\left\{-1,1,5,9,15\right\} \\ c)f(S)=\left\{0,1\right\} \\ d)f\left(S\right)=\left\{0,1,5,16\right\} \end{gathered}$$

Step-by-step solution

Step: 1 

$$\begin{gathered} \text{GivenS=}\left\{-1,0,2,4,7\right\} \\ \mathrm{a})\mathrm{f}(\mathrm{x})=1 \\ \mathrm{Then}\mathrm{f}(\mathrm{S})=\left\{-1,0,2,4,7\right\} \end{gathered}$$

Step: 2

$$\begin{gathered} \text{b)f(x)=2x+1} \\ \text{Ifx=-1,thenf(x)=-1} \\ \text{Ifx=0thenf(x)=0} \\ \text{Ifx=2thenf(x)=2} \\ \text{Ifx=4thenf(x)=4} \\ \text{Ifx=6thenf(x)=7} \\ \text{Thenf(S)=}\left\{-1,1,5,9,15\right\} \end{gathered}$$

Step: 3

$$\begin{gathered} \text{c)f(x)=}⌈x/5⌉ \\ \text{Ifx=-1,thenf(x)=0} \\ \text{Ifx=0thenf(x)=0} \\ \text{Ifx=2thenf(x)=0} \\ \text{Ifx=4thenf(x)=0} \\ \text{Ifx=7thenf(x)=1} \\ \text{Thenf(S)=}\left\{0,1\right\} \end{gathered}$$

Step :4

$$\begin{gathered} \text{d)f(x)=}⌈(x^2+1)/3⌉ \\ \text{Ifx=-1,thenf(x)=0} \\ \text{Ifx=0thenf(x)=0} \\ \text{Ifx=2thenf(x)=1} \\ \text{Ifx=4thenf(x)=5} \\ \text{Ifx=7thenf(x)=16} \\ \text{Thenf(S)=}\left\{0,1,5,16\right\} \end{gathered}$$