This answer key provides solutions for a hypothetical Algebra 2 worksheet on domain and range. Since I don't have access to your specific worksheet, I'll create example problems with varying difficulty levels, showcasing different functions and methods for finding domain and range. You can use this as a template to check your own answers. Remember to always show your work!
Instructions: Find the domain and range of each function. Express your answers in interval notation.
Problem 1: f(x) = 2x + 5
Solution:
- Domain: This is a linear function. Linear functions have a domain of all real numbers. Therefore, the domain is (-∞, ∞).
- Range: Linear functions also have a range of all real numbers. The range is (-∞, ∞).
Problem 2: g(x) = x² - 4
Solution:
- Domain: This is a quadratic function. Quadratic functions have a domain of all real numbers. Domain: (-∞, ∞).
- Range: The vertex of this parabola is (0, -4), and it opens upwards. Therefore, the range includes all values greater than or equal to -4. Range: [-4, ∞).
Problem 3: h(x) = √(x + 3)
Solution:
- Domain: The square root function is only defined for non-negative values inside the radical. Therefore, we need x + 3 ≥ 0, which means x ≥ -3. Domain: [-3, ∞).
- Range: Since the square root of a non-negative number is always non-negative, the range is all non-negative real numbers. Range: [0, ∞).
Problem 4: k(x) = 1/(x - 2)
Solution:
- Domain: This is a rational function. Rational functions are undefined when the denominator is zero. Therefore, x - 2 ≠ 0, which means x ≠ 2. Domain: (-∞, 2) U (2, ∞).
- Range: A rational function of this form (linear over linear) will never equal zero. The horizontal asymptote is y = 0. Thus, the range is all real numbers except 0. Range: (-∞, 0) U (0, ∞).
Problem 5: j(x) = |x| + 1
Solution:
- Domain: This is an absolute value function. Absolute value functions have a domain of all real numbers. Domain: (-∞, ∞).
- Range: The absolute value of any number is always non-negative. The smallest value of |x| is 0, so the smallest value of |x| + 1 is 1. Range: [1, ∞).
Problem 6 (Challenge): m(x) = √(9 - x²)
Solution:
- Domain: This is a semi-circle. We require 9 - x² ≥ 0, which means x² ≤ 9. This inequality is satisfied when -3 ≤ x ≤ 3. Domain: [-3, 3].
- Range: The largest value of √(9 - x²) occurs when x = 0, resulting in √9 = 3. The smallest value is 0, which occurs at x = ±3. Range: [0, 3].
Remember: These are example problems. Your worksheet will have different functions. Apply the same principles of identifying restrictions on the input (domain) and the resulting output values (range) to find the answers for your specific problems. If you encounter a function type not covered here (e.g., piecewise functions, logarithmic functions, etc.), refer to your textbook or class notes for guidance.
