Welcome to our exploration of domain and range. In mathematics, every function has two fundamental characteristics. The domain is the set of all possible input values, or x-values, for which the function is defined. The range is the set of all possible output values, or y-values, that the function can produce. Let's visualize this with a simple quadratic function.
Now let's explore how to find the domain of a function. The domain consists of all x-values where the function produces a real output. Different types of functions have different domain restrictions. Polynomials are defined for all real numbers. Square root functions require non-negative inputs. Fractions cannot have zero denominators. And logarithmic functions need positive inputs. Here we see the square root function, which has domain x greater than or equal to zero.
Now let's learn how to find the range of a function. The range consists of all y-values that the function can actually produce. To find the range, we can graph the function and observe the y-values it covers. We look for minimum and maximum values, consider the function's behavior at extremes, and check for horizontal asymptotes. Here we see a parabola shifted up by one unit. Its vertex is at y equals one, which is the minimum value. Since the parabola opens upward, the range is y greater than or equal to one.
Let's compare domain and range for different types of functions. The quadratic function x squared has domain of all real numbers and range of y greater than or equal to zero. The square root function has domain x greater than or equal to zero and range y greater than or equal to zero. The reciprocal function one over x has domain x not equal to zero and range y not equal to zero. Each function type has its own characteristic domain and range restrictions based on its mathematical properties.
To summarize what we have learned about domain and range: Domain represents all valid input values for a function, while range represents all possible output values a function can produce. Different function types have characteristic domain and range patterns that we can identify. Understanding domain and range is essential for proper function analysis in mathematics.