For every value of `x` the function `f(x)=1/(5^(x))` is

Options

A. Decreasing
B. Increasing
C. Neither increasing nor decreasing
D. increasin for `x gt0` and decreasing for `xlt0`

Explanation

To determine the behavior of the function \( f(x) = \frac{1}{5^x} \) for every value of \( x \), we will analyze its derivative \( f'(x) \) to check if the function is increasing or decreasing.

### Step 1: Rewrite the function
We can rewrite the function in a more convenient form:
\[ f(x) = 5^{-x} \]

### Step 2: Differentiate the function
Now, we will find the derivative \( f'(x) \) using the power rule. The derivative of \( a^x \) is given by \( a^x \ln(a) \). Therefore, for \( f(x) = 5^{-x} \):
\[ f'(x) = -5^{-x} \ln(5) \]

### Step 3: Analyze the sign of the derivative
The term \( 5^{-x} \) is always positive for all real numbers \( x \) because it is an exponential function. The natural logarithm \( \ln(5) \) is also positive since \( 5 > 1 \). Therefore, \( -5^{-x} \ln(5) \) will always be negative:
\[ f'(x) < 0 \quad \text{for all } x \]

### Step 4: Conclusion
Since the derivative \( f'(x) \) is negative for all \( x \), this means that the function \( f(x) = \frac{1}{5^x} \) is decreasing for every value of \( x \).

### Final Answer
The function \( f(x) = \frac{1}{5^x} \) is decreasing for every value of \( x \).

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