To determine the domain and range of the function [tex]\( f(x) = \frac{7}{2} \cdot 2^{x-2} - 5 \)[/tex], let's break it down step by step.
### Step 1: Determine the Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
For the given function [tex]\( f(x) = \frac{7}{2} \cdot 2^{x-2} - 5 \)[/tex]:
- The exponential expression [tex]\( 2^{x-2} \)[/tex] is defined for all real numbers.
- There are no restrictions such as division by zero or taking the logarithm of a negative number.
Therefore, the domain of [tex]\( f(x) \)[/tex] is all real numbers. In interval notation, the domain is:
[tex]\[ \text{Domain} = (-\infty, \infty) \][/tex]
### Step 2: Determine the Range
The range of a function is the set of all possible output values (y-values) that the function can take.
For the given function [tex]\( f(x) = \frac{7}{2} \cdot 2^{x-2} - 5 \)[/tex]:
- The term [tex]\( \frac{7}{2} \cdot 2^{x-2} \)[/tex] represents an exponential function multiplied by a constant.
- Exponential functions [tex]\( 2^{x-2} \)[/tex] grow rapidly for increasing x and decay towards zero for decreasing x.
- When multiplied by [tex]\( \frac{7}{2} \)[/tex], this term will always be positive and will approach zero, but never become negative.
- Subtracting 5 from this term shifts the function downward by 5 units.
Thus:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( 2^{x-2} \)[/tex] approaches zero, making [tex]\( f(x) \)[/tex] approach [tex]\( -5 \)[/tex].
- As [tex]\( x \to \infty \)[/tex], [tex]\( 2^{x-2} \)[/tex] grows very large, making [tex]\( f(x) \to \infty \)[/tex].
Therefore, the function will take on all values greater than [tex]\( -5 \)[/tex], but never actually reach [tex]\( -5 \)[/tex].
In interval notation, the range is:
[tex]\[ \text{Range} = (-5, \infty) \][/tex]
### Conclusion
Combining the results, the domain and range of the function [tex]\( f(x) = \frac{7}{2} \cdot 2^{x-2} - 5 \)[/tex] are:
- Domain: [tex]\((- \infty, \infty)\)[/tex]
- Range: [tex]\((-5, \infty)\)[/tex]
