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Algebra I

Domain and Range

Lesson

Every function has two sets to keep track of:

  • Domain: the set of valid inputs (x-values).
  • Range: the set of possible outputs (y-values).

Two big domain rules

  • Denominators can’t be zero. For f(x)=1x−3f(x) = \frac{1}{x - 3}, the domain excludes x=3x = 3.
  • Square roots can’t be negative (in real numbers). For f(x)=x−4f(x) = \sqrt{x - 4}, we need x−4≥0x - 4 \ge 0, so x≥4x \ge 4.

Worked example 1 — rational

f(x)=12x−6f(x) = \frac{1}{2x - 6}

Set the denominator equal to zero and solve: 2x−6=0⇒x=32x - 6 = 0 \Rightarrow x = 3. So x = 3 is excluded; everything else works.

Worked example 2 — root

f(x)=x+2f(x) = \sqrt{x + 2}

Inside the root must be ≥0\ge 0: x+2≥0⇒x≥−2x + 2 \ge 0 \Rightarrow x \ge -2. Smallest valid x is −2-2.

Range of a parabola

y=x2+ky = x^2 + k has a smallest value at the vertex. Since x2≥0x^2 \ge 0, the minimum is kk. Range: y≥ky \ge k.

For y=−x2+ky = -x^2 + k, the parabola opens DOWNward, so kk is the maximum.

Worked example 3 — discrete function

f={(1,4),(3,7),(5,10)}f = \{(1, 4), (3, 7), (5, 10)\}.

Domain: {1,3,5}\{1, 3, 5\}. Range: {4,7,10}\{4, 7, 10\}.

How to type your answer

For excluded values or set listings, separate with commas: 2,-5. For a minimum or maximum value, type just the number.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

For f(x)=1x−3, which x is excluded?\text{For } f(x) = \frac{1}{x - 3}, \text{ which } x \text{ is excluded?}

Problem 2

For f(x)=1x+5, which x is excluded?\text{For } f(x) = \frac{1}{x + 5}, \text{ which } x \text{ is excluded?}

Problem 3

For f(x)=x−4, smallest valid x?\text{For } f(x) = \sqrt{x - 4}, \text{ smallest valid } x?

Problem 4

f={(1,2),(3,5),(7,9)}; list the domainf = \{(1, 2), (3, 5), (7, 9)\}; \text{ list the domain}

Practice

Standard problems matching the lesson.

Problem 5

For f(x)=1x−2, which x is excluded?\text{For } f(x) = \frac{1}{x - 2}, \text{ which } x \text{ is excluded?}

Problem 6

For f(x)=1x, which x is excluded?\text{For } f(x) = \frac{1}{x}, \text{ which } x \text{ is excluded?}

Problem 7

For f(x)=12x−6, which x is excluded?\text{For } f(x) = \frac{1}{2x - 6}, \text{ which } x \text{ is excluded?}

Problem 8

For f(x)=x, smallest valid x?\text{For } f(x) = \sqrt{x}, \text{ smallest valid } x?

Problem 9

For f(x)=x−9, smallest valid x?\text{For } f(x) = \sqrt{x - 9}, \text{ smallest valid } x?

Problem 10

For f(x)=x+2, smallest valid x?\text{For } f(x) = \sqrt{x + 2}, \text{ smallest valid } x?

Problem 11

For f(x)=x2+3, minimum y?\text{For } f(x) = x^2 + 3, \text{ minimum } y?

Problem 12

For f(x)=x2−5, minimum y?\text{For } f(x) = x^2 - 5, \text{ minimum } y?

Problem 13

For f(x)=−x2+4, maximum y?\text{For } f(x) = -x^2 + 4, \text{ maximum } y?

Problem 14

f={(2,4),(4,8),(6,12)}; list the rangef = \{(2, 4), (4, 8), (6, 12)\}; \text{ list the range}

Problem 15

f={(0,1),(1,3),(2,5)}; list the domainf = \{(0, 1), (1, 3), (2, 5)\}; \text{ list the domain}

Problem 16

For f(x)=1x2−4, which x values are excluded?\text{For } f(x) = \frac{1}{x^2 - 4}, \text{ which } x \text{ values are excluded?}

Problem 17

Speed v = d/t. Which value of t is excluded?

Problem 18

Diver depth f(t) = 5*sqrt(t). Smallest valid t?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

For f(x)=1(x−2)(x+5), which x values are excluded?\text{For } f(x) = \frac{1}{(x - 2)(x + 5)}, \text{ which } x \text{ values are excluded?}

Problem 20

For f(x)=2x−6, smallest valid x?\text{For } f(x) = \sqrt{2x - 6}, \text{ smallest valid } x?

Problem 21

For f(x)=(x+1)2, minimum y?\text{For } f(x) = (x + 1)^2, \text{ minimum } y?

Problem 22

For f(x)=(x−3)2+5, minimum y?\text{For } f(x) = (x - 3)^2 + 5, \text{ minimum } y?

Problem 23

For f(x)=−(x+2)2−1, maximum y?\text{For } f(x) = -(x + 2)^2 - 1, \text{ maximum } y?

Problem 24

For f(x)=1x2−9, which x values are excluded?\text{For } f(x) = \frac{1}{x^2 - 9}, \text{ which } x \text{ values are excluded?}

Problem 25

For f(x)=16−x2, give the two boundary x values of the domain.\text{For } f(x) = \sqrt{16 - x^2}, \text{ give the two boundary } x \text{ values of the domain.}

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