Welcome! Today we'll explore what the range of a function means. The range of a function is the set of all possible output values, or y-values, that the function can produce. Think of a function as a machine that takes inputs and produces outputs. The range represents all the possible outputs this machine can create.
Let's look at some simple examples to understand range better. First, consider f of x equals x squared. Since any real number squared is always non-negative, the smallest output value is zero. The function can produce any value from zero to positive infinity, so the range is zero to infinity. Another example is the sine function, which oscillates between negative one and positive one, giving us a range of negative one to one.
To find the range of a function, follow these systematic steps. First, analyze the function type. Linear functions have a range of all real numbers, while quadratic functions require finding the vertex to determine the maximum or minimum value. For rational functions, look for horizontal asymptotes. Second, identify critical points such as maximum and minimum values, asymptotic behavior, and any domain restrictions. For example, this downward-opening parabola has a maximum at the point one comma four, giving us a range from negative infinity to four.
It's important to understand the difference between domain and range. The domain is the set of all possible input values, or x-values, that a function can accept. The range is the set of all possible output values, or y-values, that the function can produce. For example, consider the square root function. Its domain is all non-negative real numbers, since we cannot take the square root of negative numbers in the real number system. Similarly, its range is also all non-negative real numbers, because the square root of any non-negative number is always non-negative.
To summarize what we've learned about the range of functions: The range represents all possible output values a function can produce. Different types of functions have characteristic range patterns that we can identify. We find the range by analyzing critical points, maximum and minimum values, and the overall behavior of the function. Range and domain work together to completely define a function's behavior. Understanding range is essential for solving many real-world mathematical problems.