Sure, let's tackle the problem step by step.
### Step 1: Understanding the Function
The function given is [tex]\( f(x) = 4 - 2^x \)[/tex].
- The function is of the form [tex]\( a - b^x \)[/tex], where [tex]\( a = 4 \)[/tex] and [tex]\( b = 2 \)[/tex].
### Step 2: Identifying the Asymptote
An asymptote is a line that the graph of a function approaches but never touches.
- For the function [tex]\( f(x) = 4 - 2^x \)[/tex], as [tex]\( x \)[/tex] approaches infinity ( [tex]\( x \rightarrow \infty \)[/tex] ), [tex]\( 2^x \)[/tex] becomes very large, making [tex]\( 4 - 2^x \)[/tex] approach negative infinity ( [tex]\( -\infty \)[/tex] ).
- Conversely, as [tex]\( x \)[/tex] approaches negative infinity ( [tex]\( x \rightarrow -\infty \)[/tex] ), [tex]\( 2^x \)[/tex] approaches 0, making [tex]\( 4 - 2^x \)[/tex] approach 4.
Therefore, the horizontal asymptote is [tex]\( y = 4 \)[/tex].
### Step 3: Finding the Domain
The domain of a function is the set of all possible input values ( [tex]\( x \)[/tex] values) for which the function is defined.
- Since there are no restrictions on [tex]\( x \)[/tex] in [tex]\( 4 - 2^x \)[/tex], [tex]\( x \)[/tex] can be any real number.
Thus, the domain of [tex]\( f(x) = 4 - 2^x \)[/tex] is all real numbers, or [tex]\( (-\infty, \infty) \)[/tex].
### Step 4: Finding the Range
The range of a function is the set of all possible output values ( [tex]\( y \)[/tex] values ).
- For [tex]\( f(x) = 4 - 2^x \)[/tex], consider the behavior of [tex]\( 2^x \)[/tex]:
- As [tex]\( x \rightarrow \infty \)[/tex], the term [tex]\( 2^x \)[/tex] grows very large, making [tex]\( 4 - 2^x \)[/tex] very negative ( [tex]\( -\infty \)[/tex] ).
- As [tex]\( x \rightarrow -\infty \)[/tex], the term [tex]\( 2^x \)[/tex] approaches 0, making [tex]\( 4 - 2^x \)[/tex] approach 4.
Since [tex]\( 2^x \)[/tex] is always positive but never zero for all real [tex]\( x \)[/tex], [tex]\( 4 - 2^x \)[/tex] can never quite reach 4. Hence, the function values will be less than 4 but can be any value up to just below 4.
Therefore, the range of [tex]\( f(x) = 4 - 2^x \)[/tex] is [tex]\( (-\infty, 4) \)[/tex].
### Summary of Results
- Asymptote: [tex]\( y = 4 \)[/tex]
- Domain: [tex]\( (-\infty, \infty) \)[/tex]
- Range: [tex]\( (-\infty, 4) \)[/tex]
Now, you can graph [tex]\( f(x) = 4 - 2^x \)[/tex] to visualize these attributes. The graph will show the function decreasing as x increases and approaching the horizontal line [tex]\( y = 4 \)[/tex] as [tex]\( x \to -\infty \)[/tex].
