To analyze the function [tex]\( f(x) = \left( \frac{1}{4} \right)^x \)[/tex] and describe its key features, let's break down each aspect of the function step by step:
1. Behavior of the Function (Increasing/Decreasing):
- The base of the exponent is [tex]\( \frac{1}{4} \)[/tex], which is less than 1.
- For any [tex]\( x > 0 \)[/tex], the exponentiation of a fraction less than 1 results in a smaller number. Conversely, for [tex]\( x < 0 \)[/tex], this results in a number greater than 1.
- Therefore, the function is decreasing for all [tex]\( x \)[/tex].
2. Y-Intercept:
- The y-intercept occurs when [tex]\( x = 0 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
f(0) = \left( \frac{1}{4} \right)^0 = 1
\][/tex]
- The y-intercept of the function is [tex]\( y = 1 \)[/tex].
3. End Behavior:
- As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to +\infty \)[/tex]):
[tex]\[
\left( \frac{1}{4} \right)^x \to 0
\][/tex]
The function approaches 0 but never actually reaches it.
- As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]):
[tex]\[
\left( \frac{1}{4} \right)^x \to +\infty
\][/tex]
The function grows without bound.
4. Domain:
- The domain of [tex]\( f(x) \)[/tex] is the set of all real numbers because you can input any real number [tex]\( x \)[/tex] into the function.
[tex]\[
\text{Domain: } (-\infty, +\infty)
\][/tex]
5. Range:
- The range of the function is all positive real numbers. Since the base [tex]\( \frac{1}{4} \)[/tex] is a positive number and any exponentiation thereof results in another positive number.
[tex]\[
\text{Range: } (0, +\infty)
\][/tex]
To summarize the key features of the function [tex]\( f(x) = \left( \frac{1}{4} \right)^x \)[/tex]:
- The function is decreasing over its entire domain.
- The y-intercept is at [tex]\( y = 1 \)[/tex].
- As [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to 0 \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to +\infty \)[/tex].
- The domain is all real numbers [tex]\((- \infty, + \infty)\)[/tex].
- The range is [tex]\((0, +\infty) \)[/tex].
These features help to understand and sketch the graph of [tex]\( f(x) = \left( \frac{1}{4} \right)^x \)[/tex].
