Question

Question: Let $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{\lceil }{\mathbf{x}}^{\mathbf{2}}\mathbf{/}\mathbf{3}\mathbf{\rceil }$. Find $\mathit{f}\mathbf{\left(}\mathit{S}\mathbf{\right)}$if

$$\begin{gathered} \mathbf{\text{a)S={-2,-1,0,1,2,3}}} \\ \mathbf{\text{b)}}\mathbf{S}\mathbf{=\{0,1,2,3}\mathbf{,}\mathbf{4}\mathbf{,}\mathbf{5}\mathbf{\}} \\ \mathbf{\text{c)}}\mathbf{S}\mathbf{=\{}\mathbf{1}\mathbf{,}\mathbf{5}\mathbf{,}\mathbf{7}\mathbf{,}\mathbf{11}\mathbf{\}} \\ \mathbf{\text{d)}}\mathbf{S}\mathbf{=\{}\mathbf{2}\mathbf{,}\mathbf{6}\mathbf{,}\mathbf{10}\mathbf{,}\mathbf{14}\mathbf{\}} \end{gathered}$$

Short answer

Answer:

a) $f\left(s\right)=\left\{0,1,3\right\}$

b) $f\left(s\right)=\left\{0,1,3,5,8\right\}$

c) $f\left(s\right)=\left\{0,8,16,40\right\}$

d)$f\left(s\right)=\left\{1,12,33,65\right\}$

Step-by-step solution

Step: 1

Given$f\left(x\right)=⌈x^2/3⌉$

a)

If$x=-2,$ then$f\left(x\right)=⌈4/3⌉$

If$x=-1,$then$f\left(x\right)=⌈1/3⌉$

If$x=0,$then$f\left(x\right)=⌈0⌉$

If$x=1,$then$f\left(x\right)=⌈1/3⌉$

If$x=2,$then$f\left(x\right)=⌈4/3⌉$

If$x=3,$then$f\left(x\right)=⌈9/3⌉$

Therefore$f\left(s\right)=\left\{0,1,3\right\}$

Step: 2

b)

If $x=0,$then$f\left(x\right)=⌈0⌉$

If$x=1,$ then$f\left(x\right)=⌈1/3⌉$

If $x=2,$then$f\left(x\right)=⌈4/3⌉$

If $x=3,$then$f\left(x\right)=⌈9/3⌉$

If$x=4,$ then$f\left(x\right)=⌈16/3⌉$

If $x=5,$then$f\left(x\right)=⌈25/3⌉$

Therefore$f\left(s\right)=\left\{0,1,3,5,8\right\}$

Step: 3

c)

If$x=1,$ then$f\left(x\right)=⌈1/3⌉$

If$x=5,$thenrole="math" localid="1668408910780" $f\left(x\right)=⌈25/3⌉$

If$x=7,$ then$f\left(x\right)=⌈49/3⌉$

If$x=11,$ then$f\left(x\right)=⌈121/3⌉$

Therefore $f\left(s\right)=\left\{0,8,16,40\right\}$

Step: 4

d)

If$x=2,$then$f\left(x\right)=⌈4/3⌉$$f\left(x\right)=⌈4/3⌉$

If$x=6,$then$f\left(x\right)=⌈36/3⌉$

If$x=10,$then$f\left(x\right)=⌈100/3⌉$

If$x=14,$then$f\left(x\right)=⌈196/3⌉$

Therefore$f\left(s\right)=\left\{1,12,33,65\right\}$