To determine the key features of the function [tex]\( f(x) = \left( \frac{1}{4} \right)^x \)[/tex], let's analyze each aspect step-by-step:
1. Horizontal Asymptote:
- The function [tex]\( f(x) = \left( \frac{1}{4} \right)^x \)[/tex] is an exponential decay function.
- As [tex]\( x \)[/tex] approaches infinity, [tex]\( f(x) \)[/tex] approaches 0 but never actually reaches it.
- Therefore, the horizontal asymptote of the function is [tex]\( y = 0 \)[/tex].
2. Range:
- Since [tex]\( f(x) = \left( \frac{1}{4} \right)^x \)[/tex] is an exponential decay function, it produces positive values for all real [tex]\( x \)[/tex].
- The function never touches or goes below zero, and it extends to positive infinity.
- Hence, the range of [tex]\( f(x) \)[/tex] is [tex]\(\{ y \mid 0 < y < \infty \} \)[/tex].
3. Y-intercept:
- The y-intercept is the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] into the function, we get [tex]\( f(0) = \left( \frac{1}{4} \right)^0 = 1 \)[/tex].
- Therefore, the y-intercept is at the point [tex]\((0, 1)\)[/tex].
4. X-intercept:
- The x-intercept of a function is where the graph crosses the x-axis, which means [tex]\( f(x) = 0 \)[/tex].
- For [tex]\( f(x) = \left( \frac{1}{4} \right)^x \)[/tex], since it is an exponential function that only outputs positive values and never zero, there is no x-intercept.
- Therefore, the statement [tex]\( x \)[/tex]-intercept at [tex]\((3, 0)\)[/tex] is not correct.
5. Incorrect Asymptote:
- We already established the correct horizontal asymptote as [tex]\( y = 0 \)[/tex].
- The statement that there is a horizontal asymptote at [tex]\( y = 2 \)[/tex] is incorrect because the function values keep approaching zero and not 2.
6. Domain:
- The domain of exponential functions like [tex]\( f(x) = \left( \frac{1}{4} \right)^x \)[/tex] is all real numbers, i.e., [tex]\( \{ x \mid -\infty < x < \infty \} \)[/tex].
- Therefore, the domain is not restricted by any boundary like [tex]\( \{ x \mid -1 < x < \infty \} \)[/tex], making this statement incorrect.
Summarizing the correct features of the function [tex]\( f(x) = \left( \frac{1}{4} \right)^x \)[/tex]:
- Horizontal asymptote of [tex]\( y = 0 \)[/tex]
- Range of [tex]\(\{ y \mid 0 < y < \infty \} \)[/tex]
- Y-intercept at [tex]\((0, 1)\)[/tex]
The other given statements regarding an x-intercept at [tex]\((3,0)\)[/tex], a horizontal asymptote at [tex]\( y=2 \)[/tex], and a domain of [tex]\(\{ x \mid -1 < x < \infty \} \)[/tex] are not correct for this function.
