Problem 47 - Algebra and Trigonometry Real Mathematics, Real People

Question

Assume that the domain of \(f\) is the set \(A=\\{-2,-1,0,1,2\\} .\) Determine the set of ordered pairs representing the function \(f.\) $$f(x)=(x-1)^{2}$$

Step-by-Step Solution

Answer

The set of ordered pairs representing the function \(f\) is {(-2, 9), (-1, 4), (0, 1), (1, 0), (2, 1)}.

Step 1: Evaluate \(f(x)\) for \(x=-2\)

Replace \(x\) in the function \(f(x)\) with -2: \(f(-2) = ((-2) - 1)^{2} = 9 \). So, the ordered pair is (-2, 9).

Step 2: Evaluate \(f(x)\) for \(x=-1\)

Replace \(x\) in the function \(f(x)\) with -1: \(f(-1) = ((-1) - 1)^{2} = 4 \). So, the ordered pair is (-1, 4).

Step 3: Evaluate \(f(x)\) for \(x=0\)

Replace \(x\) in the function \(f(x)\) with 0: \(f(0) = ((0) - 1)^{2} = 1 \). So, the ordered pair is (0, 1).

Step 4: Evaluate \(f(x)\) for \(x=1\)

Replace \(x\) in the function \(f(x)\) with 1: \(f(1) = ((1) - 1)^{2} = 0 \). So, the ordered pair is (1, 0).

Step 5: Evaluate \(f(x)\) for \(x=2\)

Replace \(x\) in the function \(f(x)\) with 2: \(f(2) = ((2) - 1)^{2} = 1 \). So, the ordered pair is (2, 1).

Key Concepts

Ordered Pairs

When working with functions, an ordered pair is a fundamental concept that denotes a connection between two elements. In the context of a function like \( f(x) = (x-1)^2 \), each ordered pair is made up of an input (the value of \( x \)) from the domain and the corresponding output (the value of \( f(x) \)).

For instance, when you evaluate \( f(x) \) for \( x=-2 \), and find that \( f(-2) = 9 \), this results in the ordered pair (-2, 9). The first element of the pair represents the input to the function, and the second element is the output. Imagine each ordered pair as one point on a coordinate plane, where the first number corresponds to the position on the x-axis (domain), and the second number to the position on the y-axis (range).

Understanding ordered pairs is crucial because it allows you to visually interpret the behavior of functions and it forms the basis from which we can graph these relationships on a coordinate plane, letting us see the function's overall behavior.

Domain of a Function

The domain of a function is the complete set of all possible input values (commonly represented by \( x \)) for which the function is defined. In our exercise, the domain is explicitly given as the set \( A = \{-2, -1, 0, 1, 2\} \).

When you're told to evaluate a function over its domain, this means you need to find the output for every given input within the domain. This process helps establish a clear picture of how the function behaves for different values.

In the case of the quadratic function \( f(x) = (x-1)^2 \), we take each value from the set \( A \) and plug it into the function to get the corresponding output. In the textbook steps provided, this process was meticulously followed for each value within the domain, resulting in a set of ordered pairs that represent the function.

Quadratic Functions

Quadratic functions are a type of polynomial function with a degree of two. They are typically written in the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) cannot be zero. Our function, \( f(x) = (x-1)^2 \), is a quadratic function because it can be expanded to \( f(x) = x^2 - 2x + 1 \), which matches the general form.

The graph of a quadratic function is a parabola, and depending on the sign of \( a \), it opens upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)). The vertex of the parabola is the point where the graph changes direction, and in the case of \( f(x) = (x-1)^2 \), the vertex is at \( (1, 0) \) – a crucial detail that can be confirmed from the ordered pairs calculated in your exercise.

An understanding of quadratic functions is essential when learning about parabolic motion in physics, optimizing areas and volumes in calculus, and even when solving real-life problems that can be modeled using a parabola, like projectile trajectories or maximizing profit in a business scenario.