Q.The domain and range of the function $f$ given by $f(x) = 2 - |x - 5|$ is
(A) Domain $= \mathbf{R}^+$, Range $= (-\infty,\ 1]$
(B) Domain $= \mathbf{R}$, Range $= (-\infty,\ 2]$
(C) Domain $= \mathbf{R}$, Range $= (-\infty,\ 2)$
(D) Domain $= \mathbf{R}^+$, Range $= (-\infty,\ 2]$

Concept: Domain and Range of Absolute Value Function

The function $f(x) = 2 - |x - 5|$ involves an absolute value, which is defined for all real numbers.

Domain: Since $|x - 5|$ exists for every real $x$, the function $f$ is defined for all $x \in \mathbf{R}$.

Range: The absolute value $|x - 5| \geq 0$ for all $x$, with equality when $x = 5$. Therefore:

• Maximum of $f(x)$ occurs when $|x - 5| = 0$, giving $f(5) = 2 - 0 = 2$ …

The function $f(x) = 2 - |x - 5|$ combines two elementary operations: an absolute value and a linear transformation. Understanding each piece tells us where the function lives and what values it can produce.

Domain: Where can we evaluate this function?

The absolute value $|x - 5|$ is defined for every real number $x$. There are no denominators that could be zero, no square roots of negative numbers, no logarithms of non-positive values. Subtracting this absolute value from $2$ doesn't introduce any restrictions either.

The domain is all real numbers, $\mathbf{R}$.

Range: What outputs are possible?

To find the range, we need to understand the behavior of $|x - 5|$.

1. The absolute value is always non-negative: $|x - 5| \geq 0$ for all $x \in \mathbf{R}$.

2. When does it equal zero? Setting $|x - 5| = 0$ gives $x = 5$. At this point:

3. What happens as we move away from $x = 5$? For any $x \neq 5$, we have $|x - 5| > 0$, which means:

4. Can $f(x)$ become arbitrarily negative? Yes. As $x \to \infty$ or $x \to -\infty$, the quantity $|x - 5| \to \infty$, so: $f(x) = 2 - |x - 5| \to -\infty$ …