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Teacherbot 11 Sep, 13:47:08

To determine the domain and range of a function, you need to consider the set of possible input values (domain) and the set of possible output values (range). Here are some common types of functions and how to determine their domain and range:

  1. Linear Functions: A linear function is in the form f(x) = mx + b, where m and b are constants. The domain of a linear function is all real numbers (-∞, ∞) because there are no restrictions on the input values. The range depends on the slope (m) of the function. If m > 0, the range is (-∞, ∞), meaning the function covers all real numbers. If m < 0, the range is (-∞, ∞) as well. If m = 0, the range is just the y-intercept (b).

  2. Quadratic Functions: A quadratic function is in the form f(x) = ax^2 + bx + c, where a, b, and c are constants. The domain of a quadratic function is all real numbers (-∞, ∞) because there are no restrictions on the input values. The range depends on the coefficient a. If a > 0, the range is [c, ∞) or (-∞, c] depending on the vertex of the parabola. If a < 0, the range is (-∞, c] or [c, ∞) depending on the vertex.

  3. Exponential Functions: An exponential function is in the form f(x) = a^x, where a is a positive constant. The domain of an exponential function is all real numbers (-∞, ∞) because there are no restrictions on the input values. The range is (0, ∞) because the function always produces positive outputs.

  4. Logarithmic Functions: A logarithmic function is in the form f(x) = log_a(x), where a is a positive constant. The domain of a logarithmic function depends on the base (a). For a logarithmic function to be defined, the input (x) must be positive. Therefore, the domain is (0, ∞). The range is all real numbers (-∞, ∞) because logarithmic functions can produce both positive and negative outputs.

  5. Trigonometric Functions: Trigonometric functions like sine (sin), cosine (cos), and tangent (tan) have a periodic nature. The domain of these functions is all real numbers (-∞, ∞) because there are no restrictions on the input values. The range of sine and cosine functions is [-1, 1], while the range of the tangent function is all real numbers (-∞, ∞).

Remember that these are general guidelines, and there may be specific cases where the domain and range are restricted due to certain conditions or limitations.