Mathematics

Problem

The function

f:\mathbb{R}\rightarrow\mathbb{R}

is defined by

f(x)=\begin{cases}2x+1, & x<2,\\5-2x, & x\ge 2.\end{cases}

State, with brief reasoning, whether

f

is one–one.

Intuitive explanation

The key is to check whether different inputs can produce the same output—that's what one-to-one means. Notice that $f$ is made of two linear pieces with opposite slopes: the left piece increases ( $slope = 2$ ) while the right piece decreases ( $slope = -2$ ). Find the maximum and minimum values each piece can achieve, then see if their ranges overlap. If they don't, no output is repeated, and $f$ is one-to-one.

Intuitive explanation

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