Problem 1 - Precalculus with Limits

Question

A relation that assigns to each element \(x\) from a set of inputs, or ________, exactly one element \(y\) in a set of outputs, or ________, is called a ________.

Step-by-Step Solution

Answer

A relation that assigns to each element \(x\) from a set of inputs, or 'domain', exactly one element \(y\) in a set of outputs, or 'range', is called a 'function'.

Step 1: Define the sets

The first blank is referring to the set of all possible input values for the function; this set is called the 'domain'. The second blank refers to the set of all potential output values that the function can produce; this set is called the 'range'.

Step 2: Name the relationship

The third blank refers to the specific type of relationship that matches every element in the input set (the domain) to exactly one element in the output set (the range). This type of relationship is defined as a 'function'.

Key Concepts

Definition of a Function

In mathematics, a function is a rule that relates each element in one set to exactly one element in another set. This relationship is fundamental; it means that for every input (also known as an argument or independent variable), there is only one possible output (result or dependent variable).

For example, imagine a machine where you insert a number and it gives back the number doubled. If you put in '2', it gives you '4', and so forth. The machine is performing a function. It is important to note that while each input x has a single output y, multiple inputs may share the same output. For instance, both '2' and '-2' could have an output of '4' if our function squares the input number.

Domain of a Function

The domain of a function is the complete set of possible values of the independent variable. In simpler terms, it is the collection of all input values for which the function is defined and can provide an output. Think of it like the inputs that the function 'accepts'.

• If we consider a function that divides by a number, the domain would include all real numbers except for zero, because division by zero is undefined.
• For a square root function, the domain consists of all non-negative numbers since you cannot take the square root of a negative number and get a real result.

Identifying the domain is crucial because it tells us the limitations of the function and prevents us from using inputs that do not make sense.

Range of a Function

Conversely, the range of a function refers to all the possible outputs it can produce. If the domain is what you can put into a function, the range is what you can get out of it. For the machine we imagined earlier that doubles numbers, the range would be all positive even numbers if we're only inputting natural numbers.

Here's how you might determine the range:

• For a function that squares a number, any real number could be an output because squaring always gives a non-negative result.
• For a function that calculates the square root, the range would only be non-negative numbers since square roots cannot result in negative numbers.

The concept of range is fundamental in understanding how functions behave and predicting the kinds of results they can produce.