# Determine which of the following functionsf(x)=cx, g(x)=cx^{2}, h(x)=csqrt{|x|}, text{and} r(x)= frac{c}{x}

Modeling data distributions

Determine which of the following functions $$\displaystyle{f{{\left({x}\right)}}}={c}{x},\ {g{{\left({x}\right)}}}={c}{x}^{{{2}}},\ {h}{\left({x}\right)}={c}\sqrt{{{\left|{x}\right|}}},\ \text{and}\ {r}{\left({x}\right)}=\ {\frac{{{c}}}{{{x}}}}$$ can be used to model the data and determine the value of the constant c that will make the function fit the data in the table. $$\begin{array}{|c|c|} \hline x & -4 & -1 & 0 & 1 & 4 \\ \hline y & -32 & -2 & 0 & -2 & -32 \\ \hline \end {array}$$

Consider the given function and the table. Now, with the help of the values given in the table, plot a graph as, Here, it can be seen that by plotting the data, the obtained graph represents a parabola. Therefore, the function which represents the data table is $$\displaystyle{g{{\left({x}\right)}}}={c}{x}^{{{2}}}$$. The point $$\displaystyle{\left(-{4},\ -{32}\right)}$$ is on the graph. So, substitute the point, $$\displaystyle{\left(-{4},\ -{32}\right)}$$, in $$\displaystyle{g{{\left({x}\right)}}}={c}{x}^{{{2}}}$$.
$$\displaystyle-{32}={c}{\left(-{4}\right)}^{{{2}}}$$
$$\displaystyle-{32}={c}{\left({16}\right)}$$
$$\displaystyle{\frac{{-{32}}}{{{16}}}}={c}$$ Solving further, $$\displaystyle{c}=\ -{2}$$ So, $$\displaystyle{g{{\left({x}\right)}}}=\ -{2}{x}^{{{2}}}$$ Hence, the function that can be used to model the data is $$\displaystyle{g{{\left({x}\right)}}}={c}{x}^{{{2}}}$$ and the value of constant is $$\displaystyle{c}=\ -{2}$$.