Problem 34 - Introductory Algebra for College Students

Question

Express each function as a set of ordered pairs. $$f(x)=3 x+5 ; \text { domain: }\\{-1,0,1\\}$$

Step-by-Step Solution

Answer

The function \(f(x) = 3x + 5\) expressed as a set of ordered pairs for the domain \{-1,0,1\} is \{(-1,2), (0,5), (1,8)\}

Step 1: Understand The Concept of Ordered Pairs

The set of all possible ordered pairs \((x, f(x))\) produced by a function for each x-value in its domain is called the function's graph. For this exercise, the domain is explicitly mentioned, so we need to calculate the value of \(f(x)\) for each x in \{-1, 0, 1\}, which would give us the set of ordered pairs.

Step 2: Substitute The Domain Values into The Function

We calculate the value of \(f(x)\) for each x value in the domain \{-1, 0, 1\}. Let’s substitute each value of x for the function \(f(x) = 3x + 5\). For x = -1, \(f(x) = 3(-1) + 5 = 2\); for x = 0, \(f(x) = 3(0) + 5 = 5\); and for x = 1, \(f(x) = 3(1) + 5 = 8\).

Step 3: Prepare The Set of Ordered Pairs

Now, prepare the set of ordered pairs by using the results calculated in the previous step. The ordered pairs are \(-1,2\), \(0,5\), and \(1,8\).

Key Concepts

Functions

A function is a special kind of relation where each element in the domain is paired with exactly one element in the range.Think of a function like a vending machine: you input a button (in this case, an element from the domain), and the machine gives you a snack (an element from the range). The function dictates that for each specific button, you will always get the same snack.

In our exercise, the function is expressed as:

• The rule: \(f(x) = 3x + 5\) - which tells you how to transform any input \(x\).
• The domain: \(-1, 0, 1\) - representing the inputs you can use.

These components together form a complete function. The domain elements are plugged into the function's rule to produce the output, which results in ordered pairs.

Domain and Range

Understanding the domain and range of a function is like knowing the rules and possibilities of a game.

The domain encompasses all possible input values. It tells you which values can safely be used in the function without causing problems, like division by zero. In our example function \(f(x) = 3x + 5\), the domain is specifically given as \{-1, 0, 1\}.

The range consists of all the possible outputs.Once you've applied the function rule to every domain value, the results form the range. Calculating for each input, we find the function outputs 2, 5, and 8, making this the range of the function.

Graph of a Function

A graph is a visual representation of a function, often portrayed as a line or curve on a plane. For a set of ordered pairs, the graph helps us see the relationship between each input and its corresponding output.

In our exercise, the function \(f(x) = 3x + 5\) and its domain \{-1, 0, 1\} generate ordered pairs which can be graphed.

• Ordered pairs: \((-1, 2)\), \((0, 5)\), and \((1, 8)\).

Each point \((x, f(x))\) plots as a point on the graph using these pairs.

By connecting these points, or plotting them on a number line, we can visually interpret the function's behavior between the domain values. This helps us understand how inputs are related to their outputs in a concrete way.