If $x, m, n$ are integers determine whether the following are onto or one-to-one and justify.
(a) $f(x) = (x + 2)$ – One to one because no such value of $x$ has the same result
(b) $g(x) = (x^2 + 2)$ – One to one because no such value of $x$ has the same result
(c) $h(x) = (x^3 + 2)$ – One to one because no such value of $x$ has the same result
(d) $f(m, n) = m + n + 1$
– Not onto because $f(0,1)$ and $f(1,0)$ both equal to $1$
(e) $f(m, n) = |m|$
– Not onto because the negative values of $m$ will give the same result as the positive values
(f) $f(m, n) = m^2 + n^2$
– Not onto because $f(0,1)$ and $f(1,0)$ both equal to $1$
Could someone tell me if my answers are right? I am not sure whether we have to take the negative numbers as well. The questions says $n,m$ and $x$ are in $\mathbb{Z}$. So I am assuming there are both positive and negative.
Best Answer
Assuming the domain is Z, b is wrong.
The verbage use for b is weak for a,c.
Would it ask too much to actually prove it?
d, e, f are flat out wrong. Check your thinking.
To ask if a function is surjective (1-1) without stating its codomain is like asking how much water is needed to fill a glass without telling the size of the glass.