Problem 5
Question
In \(3-5 :\) a. Explain why each set of ordered pairs is or is not a function. b. List the elements of the domain. c. List the clements of the range. $$ \\{(-2,5),(-1,5),(0,5),(1,5),(2,5)\\} $$
Step-by-Step Solution
Verified Answer
This set is a function. The domain is \(-2, -1, 0, 1, 2\), and the range is \(5\).
1Step 1: Determine if it's a Function
To determine if the set of ordered pairs \((-2,5),(-1,5),(0,5),(1,5),(2,5)\) is a function, check if each input (x-value) maps to exactly one output (y-value). In this set, each distinct x-value \(-2, -1, 0, 1, 2\) maps to the same y-value \(5\). Therefore, each x-value has only one y-value, so this set is a function.
2Step 2: Identify the Domain
The domain of a set of ordered pairs consists of all the first elements (x-values) of each pair. For the set \((-2,5),(-1,5),(0,5),(1,5),(2,5)\), the domain is \(-2, -1, 0, 1, 2\).
3Step 3: Identify the Range
The range of a set of ordered pairs consists of all the second elements (y-values) of each pair. In this case, all pairs have the same y-value. Therefore, the range is just the single value \(5\).
Key Concepts
Understanding Domain and RangeExploring Ordered PairsDifference Between x-values and y-values
Understanding Domain and Range
In mathematics, the domain and range of a function are fundamental concepts that describe how the function relates input values to output values. The domain of a function refers to all the possible input values, usually represented by the variable \(x\). In the context of our exercise, the domain is made up of all the first numbers in the set of ordered pairs, which are the \(x\)-values. So, if you look at the set \((-2,5),(-1,5),(0,5),(1,5),(2,5)\), the domain is all the first numbers: \(-2, -1, 0, 1, 2\).
The range, on the other hand, is all possible output values, typically represented by the variable \(y\). It consists of the second numbers in the ordered pairs. In our example, all ordered pairs have the same \(y\)-value, which is \(5\). Thus, the range is simply the single value \(5\), no matter how many times it occurs.
Understanding these concepts is essential for grasping how functions work and for analyzing their behavior, allowing you to predict outputs for given inputs.
The range, on the other hand, is all possible output values, typically represented by the variable \(y\). It consists of the second numbers in the ordered pairs. In our example, all ordered pairs have the same \(y\)-value, which is \(5\). Thus, the range is simply the single value \(5\), no matter how many times it occurs.
Understanding these concepts is essential for grasping how functions work and for analyzing their behavior, allowing you to predict outputs for given inputs.
Exploring Ordered Pairs
Ordered pairs are a way of coupling two elements, where one element is associated with another in a sequence. In a math context, an ordered pair is written as \((x, y)\), where \(x\) is the input and \(y\) is the output. The order is crucial because \((x, y)\) is different from \((y, x)\). This distinction helps in identifying which element is considered first (the domain) and which is second (the range).
In our exercise, the ordered pairs \((-2,5),(-1,5),(0,5),(1,5),(2,5)\) mean that for each input \(x\), the corresponding output is always \(5\). They are the building blocks for understanding functions as they tell us how inputs are transformed into outputs. As such, a function can be visualized as a matching game where you pair each input with its respective output.
In our exercise, the ordered pairs \((-2,5),(-1,5),(0,5),(1,5),(2,5)\) mean that for each input \(x\), the corresponding output is always \(5\). They are the building blocks for understanding functions as they tell us how inputs are transformed into outputs. As such, a function can be visualized as a matching game where you pair each input with its respective output.
- First component: \(x\)-value (domain)
- Second component: \(y\)-value (range)
Difference Between x-values and y-values
In a set of ordered pairs, \(x\)-values and \(y\)-values have different roles, helping you understand how inputs and outputs relate within a function. The \(x\)-values are the domain, representing the inputs of the function. In our exercise, the \(x\)-values are \(-2, -1, 0, 1, 2\). These are the values you plug into the function to see what comes out.
The \(y\)-values make up the range, representing the outputs. From our ordered pairs \((-2,5),(-1,5),(0,5),(1,5),(2,5)\), the \(y\)-value is consistent: it’s always \(5\) in this case. The uniqueness of the \(y\)-value for each \(x\)-value establishes whether the set of pairs is a function.
This distinct separation helps in understanding how functions operate. By looking at \(x\)-values, you know the function inputs, and with \(y\)-values, you see how these inputs transform into outputs, showing a clear picture of the function's behavior.
The \(y\)-values make up the range, representing the outputs. From our ordered pairs \((-2,5),(-1,5),(0,5),(1,5),(2,5)\), the \(y\)-value is consistent: it’s always \(5\) in this case. The uniqueness of the \(y\)-value for each \(x\)-value establishes whether the set of pairs is a function.
This distinct separation helps in understanding how functions operate. By looking at \(x\)-values, you know the function inputs, and with \(y\)-values, you see how these inputs transform into outputs, showing a clear picture of the function's behavior.
Other exercises in this chapter
Problem 5
In \(3-6,\) find the coordinates of the ordered pair with the smallest value of \(y\) for each function. $$ f(x)=\left|\frac{x}{2}-7\right| $$
View solution Problem 5
In \(3-8,\) for each function: a. Write an expression for \(f(x) .\) b. Find \(f(5)\) \(\mathrm{f} : x \rightarrow|3 x-7|\)
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In \(6-12,\) tell whether the variables vary directly, inversely, or neither. On his way to work, Randy travels 10 miles at \(r\) miles per hour for \(h\) hours
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In \(3-10\) , the coordinates of point \(P\) on the circle with center at \(C\) are given. Write an equation of each circle: a. in center-radius form b. in stan
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