To determine the domain and range of the function [tex]\( f(x) = 5^{x-3} + 1 \)[/tex], let's go through the necessary steps:
### Domain:
The domain of a function is the set of all possible input values (x-values) that the function can accept.
1. In the function [tex]\( f(x) = 5^{x-3} + 1 \)[/tex], the expression [tex]\( 5^{x-3} \)[/tex] is an exponential function.
2. Exponential functions are defined for all real numbers because you can raise 5 to any real number power.
3. Therefore, the domain of [tex]\( f(x) \)[/tex] is all real numbers.
So, the domain of [tex]\( f(x) = 5^{x-3} + 1 \)[/tex] is:
[tex]\[ \text{Domain} = \text{all real numbers} \][/tex]
### Range:
The range of a function is the set of all possible output values (y-values).
1. Consider the expression [tex]\( 5^{x-3} \)[/tex]. For any real number [tex]\( x \)[/tex], [tex]\( 5^{x-3} \)[/tex] is always positive because the base 5 is positive and any positive number raised to any power (positive or negative) will also be positive.
2. The smallest value that [tex]\( 5^{x-3} \)[/tex] can approach is 0, but it will never actually reach 0. Thus, [tex]\( 5^{x-3} > 0 \)[/tex].
3. By adding 1 to [tex]\( 5^{x-3} \)[/tex], you shift this entire function up by 1.
4. So now, instead of [tex]\( 5^{x-3} \)[/tex] being greater than 0, [tex]\( 5^{x-3} + 1 \)[/tex] will always be greater than 1.
Therefore, the range of [tex]\( f(x) = 5^{x-3} + 1 \)[/tex] is:
[tex]\[ \text{Range} = \text{all real numbers greater than 1} \][/tex]
To summarize, the domain and range of the function [tex]\( f(x) = 5^{x-3} + 1 \)[/tex] are:
- Domain: All real numbers
- Range: All real numbers greater than 1
