Term 1 · Weeks 9–10 · Functions and Their Graphs

Standard problem

Standard

What is the domain of $f(x) = \sqrt{x - 3} + 2$ ?

Show step-by-step solution

Answer: a. $x \ge 3$

A square root is defined only when the radicand is non-negative: $x - 3 \ge 0$ .
So $x \ge 3$ . The $+2$ shifts the graph up and does not affect the domain.

Common errors to watch for

(b) Sign error when solving $x - 3 \ge 0$ .
(c) $x = 3$ is allowed because $\sqrt{0} = 0$ .
(d) Confusing the square root with the cube root, which is defined for all reals.

gm11.t1.w10.c1
Sketch the graph of the following functions: linear, absolute value, quadratic, square root, or cube root with or without technology.