Certainly! Let's go through the problem step by step to find the domain and range of the function [tex]\( y = 4^{x-5} + 3 \)[/tex].
### Domain:
1. Understanding the Function: The function given is [tex]\( y = 4^{x-5} + 3 \)[/tex]. This is an exponential function of the form [tex]\( y = a^{bx+c} + d \)[/tex], where [tex]\( a = 4 \)[/tex], [tex]\( b = 1 \)[/tex], [tex]\( c = -5 \)[/tex], and [tex]\( d = 3 \)[/tex].
2. Domain of Exponential Functions: For exponential functions [tex]\( a^{bx+c} \)[/tex], there are no restrictions on the input value [tex]\( x \)[/tex]. Exponential functions are defined for all real values of [tex]\( x \)[/tex].
3. Conclusion: Therefore, the domain of [tex]\( y = 4^{x-5} + 3 \)[/tex] is all real numbers.
[tex]\[
\text{Domain} = (-\infty, \infty)
\][/tex]
### Range:
1. Behavior of [tex]\( 4^{x-5} \)[/tex]: To understand the range, let's first analyze [tex]\( 4^{x-5} \)[/tex]. Exponential functions like [tex]\( 4^{x-5} \)[/tex] are always positive (greater than 0) and approach zero as [tex]\( x \)[/tex] approaches negative infinity. As [tex]\( x \)[/tex] increases, [tex]\( 4^{x-5} \)[/tex] grows without bound.
2. Transforming the Function: Adding 3 to [tex]\( 4^{x-5} \)[/tex]:
[tex]\[
y = 4^{x-5} + 3
\][/tex]
This shifts the entire function [tex]\( 4^{x-5} \)[/tex] up by 3 units.
3. Finding the Range: The minimum value of [tex]\( 4^{x-5} \)[/tex] is just above 0 (it never actually reaches 0). Therefore, the minimum value of [tex]\( y \)[/tex] is:
[tex]\[
y_{\text{min}} = 0 + 3 = 3
\][/tex]
As [tex]\( x \)[/tex] increases, [tex]\( 4^{x-5} \)[/tex] increases without bound, so [tex]\( y \)[/tex] will also increase without bound.
4. Conclusion: Therefore, the range of [tex]\( y = 4^{x-5} + 3 \)[/tex] is all real numbers greater than 3.
[tex]\[
\text{Range} = (3, \infty)
\][/tex]
### Summary:
- The domain of the function [tex]\( y = 4^{x-5} + 3 \)[/tex] is:
[tex]\[
\text{Domain} = (-\infty, \infty)
\][/tex]
- The range of the function [tex]\( y = 4^{x-5} + 3 \)[/tex] is:
[tex]\[
\text{Range} = (3, \infty)
\][/tex]
