- The equation \(y = 2^x + 4\) defines a set of ordered pairs \((x, y)\), satisfying the definition of a relation.
- For every input \(x\), the expression \(2^x + 4\) yields a single, unique output \(y\), satisfying the definition of a function.
- Since all functions are also relations, the equation represents both a relation and a function.
- The correct statement is: \(\boxed{\text{It represents both a relation and a function.}}\)
Explanation
- Analyzing the Definition of a Relation
A relation in mathematics is defined as any set of ordered pairs \((x, y)\). In the context of an equation involving two variables, such as \(y = 2^x + 4\), we can identify a set of ordered pairs that satisfy this equality. For any value assigned to \(x\), we can compute a corresponding value for \(y\). Since this equation describes a set of points in the Cartesian plane, it inherently qualifies as a relation.
- Analyzing the Definition of a Function
A function is a specific type of relation where each element in the domain (the input \(x\)) is mapped to exactly one element in the codomain (the output \(y\)). To determine if \(y = 2^x + 4\) is a function, we must examine if each input \(x\) results in a single, well-defined output \(y\). Since the exponential expression \(2^x\) is well-defined and produces exactly one real number for any real input \(x\), adding the constant 4 to that result will also yield exactly one unique real value for \(y\). Therefore, the equation defines a function.
- Conclusion Regarding the Equation
Because every function is, by definition, a relation, and we have established that the given equation satisfies the criteria to be both a relation and a function, we conclude that it belongs to both categories. Specifically, it assigns one unique value of \(y\) to every input \(x\), and it represents a set of ordered pairs on the coordinate plane.
Examples
In data science and programming, a 'function' is a piece of code that takes an input and returns a specific output. If you have a process where entering a specific product ID (the input \(x\)) always fetches the same price (the output \(y\)), you are using a functional relationship. Understanding whether a set of inputs and outputs forms a function helps ensure consistency in data processing and predictability in software systems.