To determine the truth about the equation [tex]\( y = 2^x + 4 \)[/tex], let's analyze it step-by-step.
1. Understanding the equation:
- The given equation is [tex]\( y = 2^x + 4 \)[/tex].
- This is an exponential equation where the base of the exponent is 2 and a constant term 4 is added.
2. Definition of a Function:
- A function is a special type of relation where every input (or [tex]\( x \)[/tex]-value) has exactly one output (or [tex]\( y \)[/tex]-value).
3. Testing for Functionality:
- To test if [tex]\( y = 2^x + 4 \)[/tex] is a function, for every [tex]\( x \)[/tex] value, determine if it produces a single [tex]\( y \)[/tex] value.
- For any real number [tex]\( x \)[/tex], when plugged into the equation, there is exactly one resultant [tex]\( y \)[/tex] value, since [tex]\(\ 2^x \)[/tex] for any real number [tex]\( x \)[/tex] produces a unique value. Therefore, [tex]\( 2^x + 4 \)[/tex] will also be unique for each [tex]\( x \)[/tex].
4. Definition of a Relation:
- A relation is simply a set of ordered pairs [tex]\(( x, y )\)[/tex].
5. Testing as a Relation:
- [tex]\( y = 2^x + 4 \)[/tex] can be represented as a set of ordered pairs [tex]\(( x, 2^x + 4 )\)[/tex].
- Therefore, it satisfies the definition of a relation as well.
So, [tex]\( y = 2^x + 4 \)[/tex] is both a function and a relation because it meets the criteria for both definitions.
Hence, the correct answer is:
D. It represents both a relation and a function.
