M151Q Precalculus · Unit 1: Functions, graphs, models · on Exam 1
MFG 2.5–2.6 Absolute value, domain, and range
Study notes for Absolute value, domain, and range in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- $|x|$ is the distance from $x$ to 0, so $|x - c|$ is the distance from $x$ to $c$. Graph: a V with corner at $c$. Equations: $|u| = k$ (with $k>0$) means $u = k$ or $u = -k$. Inequalities: $|u| < k \iff -k < u < k$; $|u| > k \iff u < -k \text{ or } u > k$.
- Domain from a formula: exclude values that make a denominator zero or an even root negative; everything else is allowed. Write the answer in interval notation.
- Range is easiest from the graph: the set of heights reached. Use transformations of the basic functions, or the vertex of a parabola.
- In applications the domain is also limited by sense: lengths are positive, people are whole numbers, time starts at 0.
Key formulas
Absolute value
$$|u| = k \Rightarrow u = \pm k;\qquad |u| < k \iff -k < u < k;\qquad |u| > k \iff u < -k\ \text{or}\ u > k$$
Domain rules
$$\text{denominator} \ne 0,\qquad \text{even radicand} \ge 0$$
Worked example
Find the domain of $h(x) = \dfrac{\sqrt{6 - 2x}}{x + 1}$ and solve $|2x - 5| = 7$ and $|x + 3| < 4$.
Show the solution
Domain: $6 - 2x \ge 0 \Rightarrow x \le 3$ and $x \ne -1$: $(-\infty,-1)\cup(-1,3]$. $|2x-5| = 7 \Rightarrow 2x - 5 = \pm7 \Rightarrow x = 6$ or $x = -1$. $|x+3| < 4 \Rightarrow -4 < x+3 < 4 \Rightarrow -7 < x < 1$.
Common mistakes
- Forgetting the second case of an absolute value equation.
- Flipping the inequality for |u| > k incorrectly (it is an "or", two rays).
- Including an endpoint in the domain where the denominator is zero.
Exam tipSay "distance" out loud: |x − 3| < 2 means "x is within 2 of 3", so the answer is (1, 5) with no algebra.
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