Problem 60 - Algebra A Combined Function

Question

Use the graph of the functions below to answer Exercises 59 through 70 If \(f(-5)=-10,\) write the corresponding ordered pair.

Step-by-Step Solution

Answer

The ordered pair is \((-5, -10)\).

Step 1: Understand Function Notation

The notation \(f(x)\) refers to the value of the function \(f\) at the input \(x\). In this problem, \(f(-5) = -10\) means that when \(x = -5\), the value of \(f\) is \(-10\).

Step 2: Formulate the Ordered Pair

An ordered pair is written in the form \((x, f(x))\), where \(x\) is the input and \(f(x)\) is the output of the function. For the problem \(f(-5) = -10\), the corresponding ordered pair is \((-5, -10)\).

Step 3: Double-check Your Work

Review the notation to ensure the ordered pair \((-5, -10)\) accurately represents the input \(x = -5\) and the function output \(f(x) = -10\).

Key Concepts

Ordered Pairs

Ordered pairs are a fundamental concept in mathematics that pair two elements together. In the context of functions, ordered pairs take the form

• \((x, f(x))\)

This pairs an input value from the domain with an output from the range determined by the function.

For instance, if a function is defined as \(f(x) = x^2\), and you choose \(x = 3\), the output is \(f(3) = 9\). Thus, the ordered pair would be \((3, 9)\).

Understanding ordered pairs helps to identify specific coordinates on a graph, linking algebraic representations to visual ones. They show exactly where a function will pass through specific points on the coordinate plane.

Graphing Functions

Graphing functions is a visual means of displaying input-output relationships expressed by a function. The graph makes it easier to see the overarching behavior of the function and interpret various aspects such as intercepts, slopes, and distinct patterns.

When you graph a function, each point on the graph corresponds to an ordered pair \((x, f(x))\). For example, the graph of the function represented by the equation \(y = 2x + 3\) is composed of all points that satisfy the function, like \((0, 3)\), \((1, 5)\), and so on.

Here are some benefits of graphing functions:

• Quickly see the function's rate of change
• Identify maximum or minimum points
• Visualize the domain and range clearly

Graphing helps learners better understand the full picture of how inputs and outputs are connected within a function.

Understanding Function Input and Output

At the heart of understanding functions is grasping the input-output relationship. Functions operate like machines: you input a value (\(x\)), the function processes it, and then delivers an output (\(f(x)\)).

This concept is clearly portrayed in function notation, where \(f(x)\) symbolizes the function's output. The way to interpret notation is straightforward—replace \(x\) with the desired input. For example, with \(f(x) = 3x + 2\), if your input \(x\) is 4, you compute the output as \(f(4) = 3(4) + 2 = 14\). This gives an ordered pair of \((4, 14)\).

Visualizing input and output helps connect abstract concepts to more tangible ideas:

• The input \(x\) is where you "start," and the function processes the input to provide an output \(f(x)\)
• Ordered pairs represent this relationship clearly with a consistent format
• This process is foundational for understanding more complex functions and concepts in math

Overall, understanding input and output is essential for effectively communicating and working with functions in both academic and real-world problem-solving contexts.