Problem 97 - Algebra A Combined Function

Question

Suppose that \(y=f(x)\) and it is true that \(f(7)=50 .\) Determine whether each is true or false. An ordered-pair solution of the function is (7,50) .

Step-by-Step Solution

Answer

True, (7, 50) is an ordered-pair solution.

Step 1: Understanding Ordered Pairs

An ordered pair is a pair of elements written in a specific order, usually in the form of \((x, y)\). It represents a point in the coordinate plane, where \(x\) is the input and \(y\) is the output of a function.

Step 2: Identify Given Information

We are given the function \(y = f(x)\) and the specific point \((x, y)\) as \((7, 50)\). According to the problem statement, \(f(7) = 50\).

Step 3: Verify Ordered Pair

An ordered pair \((a, b)\) is a solution of a function if substituting \(a\) into the function results in \(b\). Here, we substitute \(x = 7\) into the function and get \(y = 50\), which matches the ordered pair \((7, 50)\).

Step 4: Conclusion

Since the function \(f(x)\) is evaluated at \(x = 7\) giving \(y = 50\), the statement that \((7, 50)\) is an ordered-pair solution to the function is true.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where you can locate points using ordered pairs like \((7, 50)\). This plane consists of two axes: the horizontal axis (commonly known as the x-axis) and the vertical axis (often referred to as the y-axis).
Each point in this plane is determined by exactly two numbers, known as coordinates. The first number in an ordered pair represents the position along the x-axis, while the second represents the position along the y-axis.
This system of coordinates allows us to visually represent and analyze functions, equations, and other mathematical relationships.

Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input. In simpler terms, it means calculating the result of a function given a particular value of the variable.
Take the function \(f(x)\), for example. Evaluating \(f(7)\) means substituting \(7\) into the function in place of \(x\) and determining what \(y\) or \(f(x)\) outputs.
In our exercise, evaluating the function at \(x = 7\) gives us \(y = 50\), which confirms that \((7, 50)\) is a valid calculation, also known as an ordered-pair solution.

Input-Output Relationship

The input-output relationship is a fundamental concept in understanding functions. It involves recognizing how changing the input, known as \(x\), affects the output \(y = f(x)\). Essentially, this relationship tells us that each unique input should map to exactly one output, ensuring consistency and predictability in mathematical functions.
For instance, if \(f(7) = 50\), then the input of \(7\) consistently yields the output of \(50\). This predictable pattern makes it easier for us to plot or analyze functions or to determine solutions.

Function Notation

Function notation is a way of representing functions that clearly defines the input variable and the output. It is usually written as \(f(x)\), where \(f\) signifies the function and \(x\) is the variable being manipulated within the function.
One of the key advantages of function notation is that it makes it easier to understand the relationship between inputs and outputs, highlighting how inputs are processed to generate outputs. For instance, knowing \(f(7) = 50\) tells us that when 7 is the input, 50 is produced as the output.

• This notation simplifies communication about functions, making it clear which variable is in use.
• It also helps to identify the connection between different calculations within a function easily.