Let's analyze the equation [tex]\( y = 2^x + 4 \)[/tex] to determine its properties.
First, understand the definitions:
- A relation is a set of ordered pairs.
- A function is a specific type of relation where each input (independent variable [tex]\( x \)[/tex]) has exactly one output (dependent variable [tex]\( y \)[/tex]).
Given the equation [tex]\( y = 2^x + 4 \)[/tex]:
1. Identify if it represents a relation:
- The equation describes a set of ordered pairs [tex]\((x, y)\)[/tex] where [tex]\( y \)[/tex] is derived from [tex]\( x \)[/tex] by applying the specified formula.
- Thus, it represents a relation, as there is a defined relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
2. Identify if it represents a function:
- Consider the definition of a function: for every value of [tex]\( x \)[/tex] (input), there should be exactly one corresponding value of [tex]\( y \)[/tex] (output).
- Looking at the equation [tex]\( y = 2^x + 4 \)[/tex], for each value of [tex]\( x \)[/tex], the expression [tex]\( 2^x \)[/tex] yields one specific value. Adding 4 to this result does not change the uniqueness of the output.
- This means for each input [tex]\( x \)[/tex], there is a unique output [tex]\( y \)[/tex].
Since the equation [tex]\( y = 2^x + 4 \)[/tex] ensures a unique output for every input [tex]\( x \)[/tex], it satisfies the definition of a function. Additionally, since any function is inherently a relation (a set of ordered pairs where each input is associated with one output), it also satisfies the definition of a relation.
Therefore, the correct answer is:
B. It represents both a relation and a function.
