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Domain, Range, and End Behavior of Functions

Use restrictions and leading terms to describe domain, range, and end behavior. Includes a worked example, printable practice, and an answer key with explanations.

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Grade 11 Math Functions OperationsDomain, Range, and End Behavior

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CCSS.MATH.CONTENT.HSF.IF.A.1CCSS.MATH.CONTENT.HSF.IF.B.4CCSS.MATH.CONTENT.HSF.BF.B.3
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Domain, Range, and End Behavior of Functions

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Learning goal and worked example

Use restrictions and leading terms to describe domain, range, and end behavior.

For p(x) = −3x⁴+x, the term −3x⁴ dominates for large |x|, so both ends go downward. This does not identify its exact maximum.

Prerequisites: powers, radical restrictions, and interval notation. A finite graph window does not bound a function’s domain.

Read each question carefully and provide your answers in the space provided. Show all your work for full credit.

1. The graph shows y = -x² + 4. The parabola continues beyond the viewing window. Identify its domain and range.

y = -x² + 4 (continues beyond the window)-5-5-4-4-3-3-2-2-1-1001122334455y = -x² + 4 (continues beyond the window)xy

Domain:  

Range:  

2. Determine the domain of the function f(x) = √(x - 3). The domain is  .

3. What is the range of the function f(x) = x² + 2? The range is  .

4. Find the domain of the rational function f(x) = 1 / (x - 5). The domain is  .

5. Describe the end behavior of the function f(x) = -2x³ + 4x² - x + 1.

As x → ∞, f(x) →  

As x → -∞, f(x) →  

6. Describe the end behavior of the function g(x) = 3x⁴ - 2x² + 5.

As x → ∞, g(x) →  

As x → -∞, g(x) →  

7. Which of the following functions has end behavior where f(x) → ∞ as x → ∞ and f(x) → -∞ as x → -∞?

a

f(x) = x²

b

f(x) = -x³

c

f(x) = x³

d

f(x) = -x²

8. Consider the function h(x) = √(x + 4) - 1.

a) Determine the domain of h(x).

b) Determine the range of h(x).

c) Describe the end behavior of h(x) as x approaches the limits of its domain.

Answer key and worked solutions

1. y=−x²+4 has domain ℝ and range (−∞,4].

2. [3,∞), since x−3≥0. 3. [2,∞), since x²≥0. 4. (−∞,5) ∪ (5,∞), since the denominator cannot be zero.

5. As x→∞, f(x)→−∞; as x→−∞, f(x)→∞. The leading term is −2x³.

6. As x→∞ and x→−∞, g(x)→∞. The leading term is 3x⁴.

7. c, f(x)=x³.

8a. Domain [−4,∞). 8b. Range [−1,∞). 8c. As x→−4 from the right, h(x)→−1 (and h(−4)=−1); as x→∞, h(x)→∞. There is no real-domain limit as x→−∞.

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