Q.Find the domain and the range of the real function $f$ defined by $f(x) = |x - 1|$.

Concept: Domain and range of an absolute value function.

The function $f(x) = |x - 1|$ is defined for all real numbers because the expression inside the absolute value, $x - 1$, exists for every $x \in \mathbb{R}$. There are no denominators, square roots of negative numbers, or logarithms that could restrict the domain. …

Why this works: Understanding absolute value as distance

The expression $|x - 1|$ represents the distance between $x$ and $1$ on the real number line. Distance is always defined (you can measure how far any real number is from $1$) and distance is never negative. These two geometric facts immediately tell us what inputs are allowed and what outputs are possible.

Finding the domain

1. Check for restrictions

The function $f(x) = |x - 1|$ involves only subtraction and the absolute-value operation. Neither of these operations imposes any restriction:

• We can subtract $1$ from any real number
• We can take the absolute value of any real number

There are no denominators that could be zero, no even roots of potentially negative quantities, no logarithms of non-positive numbers. The function is defined everywhere.

2. Conclude the domain

Since $f(x)$ is defined for every real number $x$, the domain is the entire real line:

Finding the range

3. Understand what absolute value produces

By definition, $|a| \geq 0$ for any real number $a$. This means $f(x) = |x - 1| \geq 0$ for all $x$. So the range is contained in $[0, \infty)$.

4. Check if zero is attained

Can $f(x) = 0$? Yes, when $|x - 1| = 0$, which happens precisely when $x - 1 = 0$, i.e., $x = 1$. So $f(1) = 0$ and zero is in the range.

5. Check if all positive values are attained …