State whether each of the following statements is true or false. If you decide that a statement is false, explain why this is the case.

  1. \(x^2 -5x +6 =0\ \implies\ x = 2\).

We have \(x^2 -5x +6\) factorises to \((x-2)(x-3)\), and so the equation \(x^2 -5x +6 =0\) has the solution \(x = 2\) or \(3\).

Hence the equation does NOT definitively tell us \(x\) is \(2\) (it could be \(3\)) and the statement is false.

Note that these are TRUE statements:

\(1. \quad x^2 -5x +6 =0 \implies x = 2 \quad \text{OR} \quad x=3\).

\(2. \quad x = 2 \implies x^2 -5x +6 =0\).

  1. \(f:x\rightarrow x^2\) and \(g:x\rightarrow x^3\ \implies\ fg: x\rightarrow x^5\).

Applying the two functions \(f\) and \(g\) one after another on a number \(x\) gives \[ f\left( g(x) \right) = \left( x^3 \right)^2 = x^6,\]

using the rule \(\left(x^m\right)^n = x^{mn}\). So the given statement is false.

Note that there are two values of \(x\) so that \(x^5 = x^6\), namely \(x=0\) and \(x=1\).

So if we restrict the domain in the question to the \(x\)-values \(0\) and \(1\), the statement is true.

  1. \(|x+1| < 1\ \implies\ x < 0\).

When does \(|x+1| < 1\)? This is equivalent to -1 < \(x+1\) < 1, or \(-2 < x < 0\), which is only possibly true when \(x\) is negative.

So this statement is definitely true.