Problem 59 - Algebra A Combined Function

Question

Use the graph of the functions below to answer Exercises 59 through 70 If \(f(1)=-10\), write the corresponding ordered pair.

Step-by-Step Solution

Answer

The ordered pair is \((1, -10)\).

Step 1: Identify Given Information

We are given that the function value at \( x = 1 \) is \( f(1) = -10 \).

Step 2: Understand Ordered Pair

An ordered pair is represented as \((x, y)\), where \(x\) is the input to the function and \(y\) is the output value, or \(f(x)\).

Step 3: Formulate the Ordered Pair

Using the given function value \(f(1) = -10\), substitute \(x = 1\) and \(y = -10\) into the ordered pair form \((x, y)\).

Step 4: Write the Ordered Pair

The ordered pair corresponding to \(f(1) = -10\) is \((1, -10)\).

Key Concepts

Ordered Pairs

An ordered pair is a fundamental concept in mathematics, used to represent two related numbers or values. When we talk about function graphs, ordered pairs become extremely important. Each pair consists of two elements:

• The first element, known as the "input" or "x-coordinate."
• The second element, referred to as the "output" or "y-coordinate."

These elements are captured in the form \((x, y)\). For example, consider the pair \((1, -10)\). Here, 1 is the x-coordinate, and -10 is the y-coordinate. The coordinates tell us where to place a point on a graph in the coordinate plane. Each point represents a unique combination or relationship described by a function.
Ordered pairs are valuable because they help us visualize how two quantities relate to each other in a graph. Thus, they are an essential tool for interpreting various mathematical concepts and verifying the outputs of mathematical functions.

Function Notation

Function notation is a shorthand way of linking an input directly with its output value in a mathematical function. It is typically expressed as\(f(x)\), where

• \(f\) denotes the function's name.
• \(x\) represents the variable or input.
• \(f(x)\) gives the output resulting when \(x\) is applied to the function \(f\).

In our example, we are given \(f(1) = -10\).
This tells us that when the input value is 1, the function \(f\) provides an output of -10. Function notation helps keep our mathematical expressions clean and straightforward, making it easier to perform calculations and graphing.
Another advantage of using function notation is that it helps to formalize the relationship between quantities, allowing us to calculate values efficiently. For instance, it lets us know exactly what ordered pair to plot on the coordinate plane when given a specific function value.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graph points, lines, and curves to represent equations visually. It is crucial for visualizing ordered pairs and understanding their significance in graphing functions.
The coordinate plane consists of two perpendicular number lines:

• The horizontal axis is known as the "x-axis."
• The vertical axis is called the "y-axis."

The point where these axes intersect is known as the "origin," marked as \((0, 0)\).
To graph an ordered pair on the coordinate plane, you start at the origin. First, move horizontally to the x-coordinate. Then, move vertically to the y-coordinate. Plot the point where these movements meet.
For example, to plot \((1, -10)\), you would begin at the origin, move one unit to the right along the x-axis, then 10 units down on the y-axis.
The coordinate plane not only helps us visualize relationships as described by functions but also allows us to identify and interpret patterns through the arrangement of plotted points. This is crucial in math as it aids in better understanding and problem solving.