To analyze the function \( f(x) = \left(\frac{1}{4}\right)^{x} \), we can examine a few key properties of the function:
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Exponential Decay: The base \( \frac{1}{4} \) is less than 1, which means that as \( x \) increases, the value of \( f(x) \) decreases. Thus, \( f(x) \) represents exponential decay.
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Horizontal Asymptote: The function approaches 0 as \( x \) approaches positive infinity. Therefore, \( \lim_{x \to \infty} f(x) = 0 \).
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Value at Zero: At \( x = 0 \): \[ f(0) = \left(\frac{1}{4}\right)^{0} = 1. \]
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Value for Negative \( x \): For negative values of \( x \), \( f(x) \) increases: \[ f(-1) = \left(\frac{1}{4}\right)^{-1} = 4, \] \[ f(-2) = \left(\frac{1}{4}\right)^{-2} = 16, \] and so on. The function goes to infinity as \( x \) approaches negative infinity.
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Intercept: The y-intercept occurs at \( (0, 1) \).
In summary, the true statements about the function \( f(x) = \left(\frac{1}{4}\right)^{x} \) include:
- It is a decreasing function.
- The range of the function is \( (0, \infty) \).
- The function has a horizontal asymptote at \( y = 0 \).
- It passes through the point \( (0, 1) \).
If you have specific statements you want to validate or inquire about, please provide them for further examination.
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