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To determine which statement is true for the function [tex]\( f(x) = \left(\frac{1}{5}\right)^x - 2 \)[/tex], let's analyze the function step-by-step.
### Step 1: Find the y-intercept
To find the y-intercept, we substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = \left(\frac{1}{5}\right)^0 - 2 = 1 - 2 = -1 \][/tex]
So, the y-intercept of the function is [tex]\((0, -1)\)[/tex].
### Step 2: Analyze the behavior of the function as [tex]\( x \)[/tex] increases
We need to investigate what happens to [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] increases towards infinity:
[tex]\[ \lim_{x \to \infty} \left( \left(\frac{1}{5}\right)^x - 2 \right) \][/tex]
Since [tex]\( 0 < \frac{1}{5} < 1 \)[/tex], as [tex]\( x \)[/tex] increases, [tex]\(\left(\frac{1}{5}\right)^x\)[/tex] approaches [tex]\( 0 \)[/tex]. Hence:
[tex]\[ \lim_{x \to \infty} \left(\frac{1}{5}\right)^x - 2 = 0 - 2 = -2 \][/tex]
This means that as [tex]\( x \)[/tex] increases, the value of [tex]\( f(x) \)[/tex] approaches [tex]\(-2\)[/tex].
### Step 3: Determine if the function is increasing or decreasing
Let's consider the first derivative of the function [tex]\( f(x) = \left(\frac{1}{5}\right)^x - 2 \)[/tex]:
[tex]\[ f'(x) = \frac{d}{dx} \left( \left(\frac{1}{5}\right)^x - 2 \right) \][/tex]
Using the chain rule:
[tex]\[ f'(x) = \left(\frac{1}{5}\right)^x \ln \left(\frac{1}{5}\right) \][/tex]
Since [tex]\( 0 < \frac{1}{5} < 1 \)[/tex], [tex]\(\ln \left(\frac{1}{5}\right)\)[/tex] is negative. Therefore:
[tex]\[ f'(x) = \left(\frac{1}{5}\right)^x \ln \left(\frac{1}{5}\right) < 0 \][/tex]
This indicates that the function [tex]\( f(x) \)[/tex] is always decreasing.
### Step 4: Address the given statements
Now we can evaluate each statement based on our findings:
- A. As the value of [tex]\( x \)[/tex] increases, the value of [tex]\( f(x) \)[/tex] moves toward a constant.
This is true; [tex]\( f(x) \)[/tex] approaches [tex]\(-2\)[/tex] as [tex]\( x \)[/tex] increases.
- B. The domain of the function is [tex]\( (-2, \infty) \)[/tex].
This is false; the domain of the function is actually all real numbers [tex]\( (-\infty, \infty) \)[/tex].
- C. The function has a y-intercept at [tex]\( (0,-2) \)[/tex].
This is false; the y-intercept is actually [tex]\( (0, -1) \)[/tex].
- D. The function is increasing.
This is false; our analysis of the derivative shows that the function is decreasing.
Based on the analysis:
- The correct answer is A. As the value of [tex]\( x \)[/tex] increases, the value of [tex]\( f(x) \)[/tex] moves toward a constant.
### Step 1: Find the y-intercept
To find the y-intercept, we substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = \left(\frac{1}{5}\right)^0 - 2 = 1 - 2 = -1 \][/tex]
So, the y-intercept of the function is [tex]\((0, -1)\)[/tex].
### Step 2: Analyze the behavior of the function as [tex]\( x \)[/tex] increases
We need to investigate what happens to [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] increases towards infinity:
[tex]\[ \lim_{x \to \infty} \left( \left(\frac{1}{5}\right)^x - 2 \right) \][/tex]
Since [tex]\( 0 < \frac{1}{5} < 1 \)[/tex], as [tex]\( x \)[/tex] increases, [tex]\(\left(\frac{1}{5}\right)^x\)[/tex] approaches [tex]\( 0 \)[/tex]. Hence:
[tex]\[ \lim_{x \to \infty} \left(\frac{1}{5}\right)^x - 2 = 0 - 2 = -2 \][/tex]
This means that as [tex]\( x \)[/tex] increases, the value of [tex]\( f(x) \)[/tex] approaches [tex]\(-2\)[/tex].
### Step 3: Determine if the function is increasing or decreasing
Let's consider the first derivative of the function [tex]\( f(x) = \left(\frac{1}{5}\right)^x - 2 \)[/tex]:
[tex]\[ f'(x) = \frac{d}{dx} \left( \left(\frac{1}{5}\right)^x - 2 \right) \][/tex]
Using the chain rule:
[tex]\[ f'(x) = \left(\frac{1}{5}\right)^x \ln \left(\frac{1}{5}\right) \][/tex]
Since [tex]\( 0 < \frac{1}{5} < 1 \)[/tex], [tex]\(\ln \left(\frac{1}{5}\right)\)[/tex] is negative. Therefore:
[tex]\[ f'(x) = \left(\frac{1}{5}\right)^x \ln \left(\frac{1}{5}\right) < 0 \][/tex]
This indicates that the function [tex]\( f(x) \)[/tex] is always decreasing.
### Step 4: Address the given statements
Now we can evaluate each statement based on our findings:
- A. As the value of [tex]\( x \)[/tex] increases, the value of [tex]\( f(x) \)[/tex] moves toward a constant.
This is true; [tex]\( f(x) \)[/tex] approaches [tex]\(-2\)[/tex] as [tex]\( x \)[/tex] increases.
- B. The domain of the function is [tex]\( (-2, \infty) \)[/tex].
This is false; the domain of the function is actually all real numbers [tex]\( (-\infty, \infty) \)[/tex].
- C. The function has a y-intercept at [tex]\( (0,-2) \)[/tex].
This is false; the y-intercept is actually [tex]\( (0, -1) \)[/tex].
- D. The function is increasing.
This is false; our analysis of the derivative shows that the function is decreasing.
Based on the analysis:
- The correct answer is A. As the value of [tex]\( x \)[/tex] increases, the value of [tex]\( f(x) \)[/tex] moves toward a constant.
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