To answer the question regarding the equation [tex]\( y = 2^x + 4 \)[/tex], let's analyze it in detail.
1. Understanding Relations and Functions:
- A relation is a set of ordered pairs, and a relation describes how elements from one set (inputs, usually denoted as [tex]\(x\)[/tex]) are related to elements in another set (outputs, usually denoted as [tex]\(y\)[/tex]).
- A function is a specific type of relation where each input [tex]\(x\)[/tex] is associated with exactly one output [tex]\(y\)[/tex]. In other words, for every [tex]\(x\)[/tex], there is a unique [tex]\(y\)[/tex].
2. Checking if the Equation [tex]\( y = 2^x + 4 \)[/tex] Represents a Relation:
- Since this equation pairs inputs [tex]\(x\)[/tex] with outputs [tex]\(y\)[/tex] (i.e., it can generate ordered pairs [tex]\((x, y)\)[/tex]), it certainly represents a relation.
3. Checking if the Equation [tex]\( y = 2^x + 4 \)[/tex] Represents a Function:
- For any given [tex]\(x\)[/tex], the corresponding value of [tex]\(y\)[/tex] is given by [tex]\(2^x + 4\)[/tex].
- For instance, if [tex]\(x = 1\)[/tex], then [tex]\(y = 2^1 + 4 = 6\)[/tex]. If [tex]\(x = 2\)[/tex], then [tex]\(y = 2^2 + 4 = 8\)[/tex], and so on.
- Importantly, for every unique input [tex]\(x\)[/tex], there is always a unique output [tex]\(y\)[/tex]. This satisfies the definition of a function.
4. Conclusion:
- The equation [tex]\( y = 2^x + 4 \)[/tex] satisfies both the criteria of being a relation (since it pairs [tex]\(x\)[/tex] and [tex]\(y\)[/tex]) and a function (because each [tex]\(x\)[/tex] has only one [tex]\(y\)[/tex]).
Therefore, the correct statement about the equation [tex]\( y = 2^x + 4 \)[/tex] is:
D. It represents both a relation and a function.
