Domain and range in interval notation

About this lesson

Precalculus › Functions and their graphs

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Domain and range in interval notation

Warm-up

Three quick questions from earlier lessons.

Discover

Which numbers can you put into f(x)=xf(x) = \sqrt{x}?

  1. 0 and the positive numbers
  2. Every number
  3. Only the positive numbers, not 0

Key idea

The domain is the set of inputs a function accepts. The range is the set of outputs it gives.

Leave out of the domain every xx that makes a denominator 00 or puts a negative number under a square root. A square bracket includes the end: [2,∞)[2, \infty) contains 2; a parenthesis leaves it out: (2,∞)(2, \infty) does not.

Worked example

Find the domain of f(x)=x+2f(x) = \sqrt{x + 2}.

x+2≥0x + 2 \ge 0
The inside of a square root must not be negative.
x≥−2x \ge -2
Subtract 2 from both sides.
Domain: [−2,∞)[-2, \infty)
−2 is allowed (√0 = 0), so use a square bracket.

Your turn, guided

Your turn: f(x)=x−5f(x) = \sqrt{x - 5}. Fill in the missing line.

x−5≥0x - 5 \ge 0
The inside must not be negative.
The domain starts at x=5x = 5: [5,∞)[5, \infty)
Add 5 to both sides: the domain starts at this number.

Practice ★

  1. What is the domain of f(x)=x−9f(x) = \sqrt{x - 9}? Give it as an interval.
  2. What is the domain of f(x)=x−5f(x) = \sqrt{x - 5}? Give it as an interval.
  3. What is the domain of f(x)=x−6f(x) = \sqrt{x - 6}? Give it as an interval.
  4. What is the domain of f(x)=x−4f(x) = \sqrt{x - 4}? Give it as an interval.

Watch out!

Practice ★★

  1. What is the domain of f(x)=9x−6f(x) = \dfrac{ 9 }{x - 6}? Give it as an interval or a union of intervals.
  2. What is the domain of f(x)=6x−5f(x) = \dfrac{ 6 }{x - 5}? Give it as an interval or a union of intervals.
  3. What is the domain of f(x)=4x−3f(x) = \dfrac{ 4 }{x - 3}? Give it as an interval or a union of intervals.
  1. What is the range of f(x)=(x−5)2+9f(x) = (x - 5)^2 + 9? Give it as an interval.
  2. What is the range of f(x)=(x−3)2+6f(x) = (x - 3)^2 + 6? Give it as an interval.
  3. What is the range of f(x)=(x−5)2+4f(x) = (x - 5)^2 + 4? Give it as an interval.

Exit ticket

Three questions to check you have got it.

  1. What is the range of f(x)=(x−5)2+9f(x) = (x - 5)^2 + 9? Give it as an interval.
  2. What is the range of f(x)=(x−3)2+6f(x) = (x - 3)^2 + 6? Give it as an interval.
  3. What is the range of f(x)=(x−5)2+4f(x) = (x - 5)^2 + 4? Give it as an interval.

Summary card

Domain and range

  • Domain: the inputs. Leave out zero denominators and negatives under a square root.
  • Range: the outputs.
  • [a,∞)[a, \infty) includes aa; (a,∞)(a, \infty) does not.

Watch out: the range uses yy-values, not the xx of a vertex.

End of the lesson

Next lesson: Piecewise-defined functions and absolute value Back to the chapter

Answers: Domain and range in interval notation

A

  1. [9,∞)[9, \infty)
  2. [5,∞)[5, \infty)
  3. [6,∞)[6, \infty)
  4. [4,∞)[4, \infty)

B

  1. (−∞,6)∪(6,∞)(-\infty, 6) \cup (6, \infty)
  2. (−∞,5)∪(5,∞)(-\infty, 5) \cup (5, \infty)
  3. (−∞,3)∪(3,∞)(-\infty, 3) \cup (3, \infty)

C

  1. [9,∞)[9, \infty)
  2. [6,∞)[6, \infty)
  3. [4,∞)[4, \infty)

D

  1. [9,∞)[9, \infty)
  2. [6,∞)[6, \infty)
  3. [4,∞)[4, \infty)

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