Domain and Range of Functions Explained with Examples

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Welcome to our exploration of domain and range. A function creates a relationship between input values, called the domain, and output values, called the range. The domain represents all possible x-values that we can input into the function, while the range represents all possible y-values that the function can output. Let's visualize this with a simple quadratic function. The domain represents all possible input values for a function. For a linear function like y equals half x plus one, we can substitute any real number for x and get a valid output. This means the domain is all real numbers, from negative infinity to positive infinity. The red arrows and line show that we can extend the input values infinitely in both directions. Not all functions have domains that include all real numbers. The square root function f of x equals square root of x has a restricted domain. Since we cannot take the square root of negative numbers in the real number system, the domain is restricted to x greater than or equal to zero. This is shown by the solid dot at zero and the arrow extending to positive infinity, indicating the domain is from zero to positive infinity. The range represents all possible output values of a function. For the parabola f of x equals x squared plus one, we can see that the lowest point occurs at the vertex where y equals one. Since the parabola opens upward, the function can produce any y-value greater than or equal to one, but never any value less than one. Therefore, the range is from one to positive infinity, shown by the green line with a solid dot at y equals one.