Question

# Which one of the following function has an inverse? f(x) = x^2, g(x) = x^3

Analyzing functions
Which one of the following function has an inverse?
$$\displaystyle{f{{\left({x}\right)}}}={x}^{{2}},{g{{\left({x}\right)}}}={x}^{{3}}$$

1) For $$\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}$$
$$\displaystyle{F}{\left(-{1}\right)}={\left(-{1}\right)}^{{2}}={1}{\quad\text{and}\quad}{f{{\left({1}\right)}}}={1}^{{2}}={1}$$
$$\displaystyle{x}{1}\ne{x}{2},{b}{u}{t}{f{{\left({x}{1}\right)}}}={f{{\left({x}{2}\right)}}}$$, so the function is not One-to-One, it doesn’t have an inverse.
2) For $$\displaystyle{g{{\left({x}\right)}}}={x}^{{3}}$$
$$\displaystyle{G}{\left(-{1}\right)}={\left(-{1}\right)}^{{3}}=-{1}{\quad\text{and}\quad}{g{{\left({1}\right)}}}={1}^{{3}}={1}$$
$$\displaystyle{X}{1}\ne{x}{2}{\quad\text{and}\quad}{g{{\left({x}{1}\right)}}}\ne{g{{\left({x}{2}\right)}}}$$, so function is One-to-One, it has an inverse.