Consider the following statements: 1. The relation f defined by \(f(x)= \begin{cases}x^3, & 0 \leq x \leq 2 \\ 4 x, & 2 \leq x \leq 8\end{cases}\) is a function. 2. The relation g defined by \(g(x)= \begin{cases}x^2, & 0 \leq x \leq 4 \\ 3 x, & 4 \leq x \leq 8\end{cases}\) is a function. Which of the statements given above is/are correct?
1 only
Analyzing Piecewise Relations to Determine if They Are Functions
To determine if a given relation is a function, we must ensure that for every element in its domain, there is exactly one corresponding element in its codomain. When dealing with piecewise-defined relations, a crucial point to check is the boundary where the definition changes. At this boundary point, the value obtained from both defining expressions must be the same. If the values differ, the relation is not a function because the boundary point would be mapped to multiple outputs.
Evaluating Statement 1: Relation f(x)
The first statement defines the relation \(f(x)\) as:
\(f(x)= \begin{cases}x^3, & 0 \leq x \leq 2 \\ 4 x, & 2 \leq x \leq 8\end{cases}\)
The domain of this relation is given as \(0 \leq x \leq 8\). The point where the definition changes is \(x=2\). We need to check if the two parts of the definition agree at this boundary point.
- Using the first part (\(x^3\)) for \(x=2\): \(f(2) = 2^3 = 8\)
- Using the second part (\(4x\)) for \(x=2\): \(f(2) = 4(2) = 8\)
Since the value obtained from both expressions at \(x=2\) is the same (which is 8), the relation \(f\) assigns a unique output to the point \(x=2\). For all other points within their respective intervals (\(0 \leq x < 2\) and \(2 < x \leq 8\)), \(x^3\) and \(4x\) are standard functions that assign a unique output for each input. Thus, the relation \(f\) is a function.
Statement 1 is correct.
Evaluating Statement 2: Relation g(x)
The second statement defines the relation \(g(x)\) as:
\(g(x)= \begin{cases}x^2, & 0 \leq x \leq 4 \\ 3 x, & 4 \leq x \leq 8\end{cases}\)
The domain of this relation is given as \(0 \leq x \leq 8\). The point where the definition changes is \(x=4\). We need to check if the two parts of the definition agree at this boundary point.
- Using the first part (\(x^2\)) for \(x=4\): \(g(4) = 4^2 = 16\)
- Using the second part (\(3x\)) for \(x=4\): \(g(4) = 3(4) = 12\)
The values obtained from the two expressions at \(x=4\) are different (16 and 12). This means that according to this definition, the input \(x=4\) is associated with two different outputs, 16 and 12. By the definition of a function, each input must have exactly one output. Since \(x=4\) has two outputs, the relation \(g\) is not a function.
Statement 2 is incorrect.
Conclusion on the Statements
Based on the analysis:
- Statement 1: The relation \(f(x)\) is a function. (Correct)
- Statement 2: The relation \(g(x)\) is a function. (Incorrect)
Therefore, only Statement 1 is correct.
| Statement | Relation Definition | Boundary Point Check | Value 1 | Value 2 | Result | Is it a Function? |
|---|---|---|---|---|---|---|
| 1 | \(f(x)= \begin{cases}x^3, & 0 \leq x \leq 2 \\ 4 x, & 2 \leq x \leq 8\end{cases}\) | \(x=2\) | \(2^3 = 8\) | \(4(2) = 8\) | Values Match | Yes |
| 2 | \(g(x)= \begin{cases}x^2, & 0 \leq x \leq 4 \\ 3 x, & 4 \leq x \leq 8\end{cases}\) | \(x=4\) | \(4^2 = 16\) | \(3(4) = 12\) | Values Don't Match | No |
Revision Table: Key Concepts for Functions
| Concept | Definition | Relevance to Question |
|---|---|---|
| Function | A relation where each element in the domain is associated with exactly one element in the codomain. | The core definition used to evaluate both statements. |
| Domain | The set of all possible input values (x-values) for a function. | Given intervals for x define the domain for each relation. |
| Piecewise Function | A function defined by multiple sub-functions, each applied to a different interval of the domain. | The specific type of relations being analyzed in the question. |
| Boundary Point | A point where the definition of a piecewise function changes from one expression to another. | Critical point to check for function property in piecewise definitions. |
Additional Information: Understanding Relations vs. Functions
A relation is simply a set of ordered pairs \((x, y)\). A function is a special type of relation. For a relation to be a function, it must satisfy the vertical line test: any vertical line drawn through the graph of the relation intersects the graph at most once. This is equivalent to saying that for any given x-value in the domain, there is only one corresponding y-value.
In the case of piecewise definitions, checking the boundary points is essentially applying the vertical line test at that specific x-value. If the two expressions give different y-values at the boundary x, it means the point \((x, y_1)\) and \((x, y_2)\) with \(y_1 \neq y_2\) both exist in the relation, violating the function definition. Our calculation for statement 2 showed this violation at \(x=4\) with points \((4, 16)\) and \((4, 12)\) both implied by the definition if it were a function.
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