To find the domain and range of the function [tex]\( f(x) = 4^{-x} - 1 \)[/tex], let's go through the process step by step.
### Domain:
The domain of a function consists of all the possible values of [tex]\( x \)[/tex] for which the function is defined.
For the function [tex]\( f(x) = 4^{-x} - 1 \)[/tex]:
- The term [tex]\( 4^{-x} \)[/tex] involves raising 4 to the power of [tex]\(-x\)[/tex].
- Exponents can be applied to any real number without any restriction.
- Therefore, there are no constraints on [tex]\( x \)[/tex].
So, the domain of the function is all real numbers. We can write this as:
[tex]\[ \text{Domain} = (-\infty, \infty) \][/tex]
### Range:
The range of a function is the set of all possible output values (i.e., [tex]\( f(x) \)[/tex]).
To find the range of [tex]\( f(x) = 4^{-x} - 1 \)[/tex]:
1. Consider the expression [tex]\( 4^{-x} \)[/tex]:
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( -x \)[/tex] approaches [tex]\( \infty \)[/tex]. Therefore, [tex]\( 4^{-x} \)[/tex] approaches 0 from the positive side.
- As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], [tex]\( -x \)[/tex] approaches [tex]\( -\infty \)[/tex]. Therefore, [tex]\( 4^{-x} \)[/tex] approaches [tex]\( \infty \)[/tex].
2. Subtracting 1:
- When [tex]\( 4^{-x} \)[/tex] is at its minimum value (which is greater than 0), [tex]\( 4^{-x} = 0^{+} \)[/tex]:
[tex]\[ 4^{-x} - 1 \approx 0 - 1 = -1 \][/tex]
- When [tex]\( 4^{-x} \)[/tex] is at its maximum value (which is [tex]\( \infty \)[/tex]):
[tex]\[ 4^{-x} - 1 = \infty - 1 = \infty \][/tex]
- Therefore, as [tex]\( x \)[/tex] varies over all real numbers, [tex]\( f(x) = 4^{-x} - 1 \)[/tex] will take on values from [tex]\(-1\)[/tex] to [tex]\( \infty \)[/tex].
So, the range of the function is:
[tex]\[ \text{Range} = (-1, \infty) \][/tex]
### Conclusion:
- Domain: [tex]\((-\infty, \infty)\)[/tex]
- Range: [tex]\((-1, \infty)\)[/tex]
Thus, the correct domain and range for the function [tex]\( f(x) = 4^{-x} - 1 \)[/tex] are:
[tex]\[ \text{Domain} = (-\infty, \infty) \][/tex]
[tex]\[ \text{Range} = (-1, \infty) \][/tex]
