Which of these function statements are true?

Which of these function statements are true?

Question

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. $x^2 -5x +6 =0\ \implies\ x = 2$.
2. $f:x\rightarrow x^2$ and $g:x\rightarrow x^3\ \implies\ fg: x\rightarrow x^5$.
3. $|x+1| < 1\ \implies\ x < 0$.

Solution

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:

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

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

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.

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.