To determine the domain and range of the function [tex]\( y = 4^{x-5} + 3 \)[/tex], follow these steps:
### Domain
The domain of a function represents all the possible values of the independent variable, in this case, [tex]\( x \)[/tex], for which the function is defined.
1. Start with the base function [tex]\( 4^{x-5} \)[/tex].
- Exponential functions of the form [tex]\( a^x \)[/tex] where [tex]\( a \)[/tex] is a positive real number are defined for all real numbers [tex]\( x \)[/tex].
- Subtracting 5 from [tex]\( x \)[/tex] does not change the fact that the function will be defined for all real numbers.
Thus, the domain of the function [tex]\( y = 4^{x-5} + 3 \)[/tex] is all real numbers.
### Range
The range of a function is the set of possible output values (values of [tex]\( y \)[/tex]) that the function can produce.
1. Analyze the base function [tex]\( 4^{x-5} \)[/tex].
- [tex]\( 4^{x-5} \)[/tex] is an exponential function that is always positive.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 4^{x-5} \)[/tex] approaches 0, but never reaches it.
- As [tex]\( x \)[/tex] increases, [tex]\( 4^{x-5} \)[/tex] increases rapidly without bound.
2. Now, consider the full function [tex]\( y = 4^{x-5} + 3 \)[/tex].
- Adding 3 to [tex]\( 4^{x-5} \)[/tex] shifts the entire graph of the exponential function upwards by 3 units.
- Therefore, [tex]\( y \)[/tex] will always be greater than 3, because [tex]\( 4^{x-5} \)[/tex] is always positive.
Thus, the range of the function [tex]\( y = 4^{x-5} + 3 \)[/tex] is [tex]\( y > 3 \)[/tex].
### Summary
- The domain of the function is: all real numbers.
- The range of the function is: [tex]\( y > 3 \)[/tex].
So, the domain of this function is
[tex]\[ \text{all real numbers} \][/tex]
And the range of this function is
[tex]\[ y > 3 \][/tex]
