8. Five marks exercise problems II

1. Show that the function \(f: \mathbb{N} \rightarrow \mathbb{N}\) defined by \(f(m) = m^2 + m + 3\) is one-to-one function.

 

 

2. In each of the following cases, state whether the function is bijective or not. Justify your answer.

 

(i) \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x) = 2x + 1\)

 

(ii) \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x) = 3 - 4x^2\)

 

 

3. The distance \(S\) an object travels under the influence of gravity in time \(t\) seconds is given by \(S(t) = \frac{1}{2} gt^2 + at + b\) where (\(g\)) is the acceleration due to gravity), \(a\), \(b\) are constants. Check if the function \(S(t)\) is one-to-one.