What is the difference between domain and range?
The domain of a function is the set of all values for the independent variable (in the following example, that's the x) for which the function is defined. That is, if you have a function such as
f(x) = 1/x
Then this function is undefined at x=0, so the domain would be all real numbers except 0.
As another example, suppose you had the function
f(x) = sqrt(x).
We know that the the square root of a negative number is not a real number, so this function isn't defined when x is negative. So, the domain is all real numbers greater than or equal to 0.
The range, on the other hand is the set of all values that the function can take, given the domain. So, for the first example, the range is all real numbers. That is, any real number can be obtained from this function if I plug in the right value from the domain.
For the second example, the square root of a number is always positive, so the range is all real numbers greater or equal to 0.
The strategy for determining the domain and the range is this:
1. Start with the domain. See if there are any numbers that need to be excluded. For the most part, you're looking for x's in the denominator of a fraction or x's under a square root.
2. Once you figure out the domain, then try to reason out what values the function can take on given the domain that you just figured out.
• Describe a real-life situation that could be modeled by a function.
The conversion from fahrenheit to celsius is given by a function (it's even linear). f(C) = 5/9 F - 32
Describe the values for x that may not be appropriate values even when they are defined by your classmates' function.
The denominator of a fraction cannot be 0, so any time x occurs in the denominator, we must be wary of that. Furthermore, where there is a square root symbol, you cannot take the square root of a negative number, so we must be careful for that as well.
A function could, for example, indicate the amount of bone strength (y) in a living body over time in years(x). It would not make sense to look at negative years, because the person would not yet be born. Likewise, looking beyond 100 years might not make sense, as many people do not live to be 100.
These are excellent examples of values of x that would be inappropriate as well
To determine the domain and range of a function, you generally need to consider any restrictions on the values that x can take and then examine the resulting values of y. Here is a step-by-step process to determine the domain and range of a function:
1. Start by analyzing the domain: Look for any values of x that might cause the function to be undefined. These can include values such as division by zero, square roots of negative numbers, or logarithms of non-positive numbers. Identify the restrictions on x and exclude those values from the domain.
2. After determining the domain, examine the resulting values of y. These values represent the output or range of the function. Consider any limitations or patterns in the values y can take.
Now, let's consider a real-life situation that can be modeled by a function:
One example is determining the distance traveled by a car (y) based on the time spent driving (x). In this scenario, the function could be linear, such as y = mx + b, where m is the car's speed in miles per hour and b is the initial position. The time spent driving cannot be negative, so the domain would be x ≥ 0. The range will depend on the car's speed and the amount of time driven.
However, it is essential to consult your classmates' specific function to determine values of x that may not be appropriate. For example, if their function contains a square root or a division by x, you need to avoid values that would make those operations undefined (e.g., x = 0 for division or negative x for square root).
Additionally, in real-life scenarios, there may be practical limitations to consider. For instance, negative values of x may not make sense in a process that involves time, or extremely large values of x may not be meaningful due to practical restrictions or human lifespan.
In summary, the domain and range of a function can be determined by analyzing any restrictions on the values of x and observing the resulting values of y. Real-life situations often have additional practical limitations to consider when determining appropriate values for x.
- For the Fahrenheit to Celsius conversion function, it would not make sense to input temperatures below absolute zero (-273.15 degrees Celsius) because it is not physically possible for a temperature to be lower than that.
- For the function modeling bone strength over time, negative years or years beyond a person's maximum lifespan (e.g., 150 years) would be inappropriate values as they do not correspond to the real-world context being modeled.
In general, it's important to consider the practical limitations and constraints of the real-world situation when determining the appropriate values for x in a function.