Let's analyze the function [tex]\( f(x) = \left(\frac{1}{4}\right)^x \)[/tex] and its key features step-by-step.
### 1. Range of the function
To determine the range, we need to understand the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] varies.
- As [tex]\( x \to \infty \)[/tex], the expression [tex]\( \left(\frac{1}{4}\right)^x \)[/tex] approaches 0 because [tex]\( \frac{1}{4} \)[/tex] is a positive fraction less than 1.
- As [tex]\( x \to -\infty \)[/tex], the expression [tex]\( \left(\frac{1}{4}\right)^x \)[/tex] grows larger without bound (approaching [tex]\( \infty \)[/tex]).
- Since [tex]\( \left(\frac{1}{4}\right)^x \)[/tex] is always positive for all real [tex]\( x \)[/tex], it never equals zero or negative.
Thus, the range of the function [tex]\( f(x) = \left(\frac{1}{4}\right)^x \)[/tex] is [tex]\((0, \infty)\)[/tex], meaning [tex]\( 0 < y < \infty \)[/tex].
### 2. Horizontal asymptote
The horizontal asymptote of a function describes the value that the function approaches as [tex]\( x \)[/tex] tends to [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex].
- As [tex]\( x \to \infty \)[/tex], [tex]\( \left(\frac{1}{4}\right)^x \to 0 \)[/tex].
Hence, the horizontal asymptote of [tex]\( f(x) \)[/tex] is [tex]\( y = 0 \)[/tex].
### 3. Domain of the function
The domain of a function is the complete set of possible values of the independent variable [tex]\( x \)[/tex].
- The function [tex]\( \left(\frac{1}{4}\right)^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
Therefore, the domain of [tex]\( f(x) = \left(\frac{1}{4}\right)^x \)[/tex] is [tex]\(\{x \mid -\infty < x < \infty \}\)[/tex].
### 4. [tex]\( y \)[/tex]-intercept
The [tex]\( y \)[/tex]-intercept occurs when [tex]\( x = 0 \)[/tex]. We substitute [tex]\( x = 0 \)[/tex] into the function to find the corresponding [tex]\( y \)[/tex]-value:
[tex]\[ f(0) = \left(\frac{1}{4}\right)^0 = 1 \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept is [tex]\( (0, 1) \)[/tex].
### Conclusion
Summarizing the key features of the function [tex]\( f(x) = \left(\frac{1}{4}\right)^x \)[/tex]:
- The range of [tex]\( f(x) \)[/tex] is [tex]\(\{ y \mid 0 < y < \infty \}\)[/tex].
- The horizontal asymptote is [tex]\( y = 0 \)[/tex].
- The domain of [tex]\( f(x) \)[/tex] is [tex]\( \{ x \mid -\infty < x < \infty \} \)[/tex].
- The [tex]\( y \)[/tex]-intercept is at [tex]\( (0, 1) \)[/tex].
Thus, the statements that describe the key features of the function [tex]\( f(x) = \left(\frac{1}{4}\right)^x \)[/tex] are:
- Range of [tex]\( \{ y \mid 0 < y < \infty \} \)[/tex]
- Horizontal asymptote of [tex]\( y = 0 \)[/tex]
- Domain of [tex]\( \{ x \mid -\infty < x < \infty \} \)[/tex] (corrected from [tex]\(\{x \mid -1- [tex]\( y \)[/tex]-intercept at [tex]\( (0, 1) \)[/tex]
It should be noted that there was an error in the original statement regarding the domain, which should be corrected to [tex]\(-\infty < x < \infty\)[/tex].
