What is the difference between domain and range? Describe a real-life situation that could be modeled by a function
A function generates a number (output) if we supply an input value. For example, if f(x) = x², then x is the input value, and f(x) or x² is the output value.
The set of all possible input values is the domain. In the case of f(x)=x², the domain are all real numbers, or dom(f(x))=ℝ.
Range is the set of all possible output values, and for f(x)=², it is limited to the non-negative real numbers, or [0,∞).
A real life situation could be the cost of gasoline, given by f(x)=price*x where price is the price per gallon/litre, and x is the number of units purchased.
The domain and range are important concepts in mathematics, particularly when working with functions. Let's start with the definitions:
- Domain: The domain of a function is the set of all possible input values or values for the independent variable. In simpler terms, it is the set of x-values for which the function is defined.
- Range: The range of a function is the set of all possible output values or values for the dependent variable. In simpler terms, it is the set of y-values that the function can produce.
To understand the difference between domain and range, let's consider a real-life situation that could be modeled by a function:
Example: The height of a ball thrown in the air
Suppose you throw a ball in the air, and you want to model its height as a function of time. In this case, you can define a function, let's call it "h(t)", where "h" represents the height and "t" represents time.
- Domain: The domain of this function will be all the possible values for time that make sense in this context. It will typically be a subset of the real numbers, starting from 0 or the time at which you throw the ball and continuing until the ball hits the ground or reaches its maximum height and falls back down.
- Range: The range of this function will be all the possible values for height that the ball can reach during its flight. It will typically be a subset of the real numbers as well, depending on factors such as the initial velocity of the throw, air resistance, and gravitational force.
To summarize, the domain represents all the valid values for the independent variable (in this case, time), and the range represents the corresponding values for the dependent variable (in this case, height). The domain and range are essential in understanding the behavior and limitations of a function.