To determine which statements about the function [tex]\( f(x) = 3(0.95)^x - 5 \)[/tex] are true, let's analyze the function step-by-step.
### Step 1: Identify the Horizontal Asymptote
First, let's consider the long-term behavior of the function as [tex]\( x \)[/tex] approaches infinity. Notice that [tex]\( 0.95 < 1 \)[/tex] and thus [tex]\( (0.95)^x \)[/tex] approaches 0 as [tex]\( x \)[/tex] approaches infinity.
[tex]\[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} [3(0.95)^x - 5] = 3 \cdot 0 - 5 = -5 \][/tex]
So, the function has a horizontal asymptote at [tex]\( y = -5 \)[/tex].
### Step 2: Check if the Function is Decreasing or Increasing
Let's examine how [tex]\( f(x) = 3(0.95)^x - 5 \)[/tex] behaves as [tex]\( x \)[/tex] increases. The base of the exponent, [tex]\( 0.95 \)[/tex], is less than 1. Therefore, [tex]\( (0.95)^x \)[/tex] is a decreasing function, meaning as [tex]\( x \)[/tex] increases, [tex]\( (0.95)^x \)[/tex] decreases.
- Since [tex]\( f(x) \)[/tex] can be written as a multiple of a decreasing function minus a constant, [tex]\( 3(0.95)^x \)[/tex] is also a decreasing function.
- Subtracting 5 from a decreasing function [tex]\( 3(0.95)^x \)[/tex] doesn't change its monotonicity.
Therefore, [tex]\( f(x) = 3(0.95)^x - 5 \)[/tex] is a decreasing function.
### Step 3: Determine the Range of the Function
Considering [tex]\( f(x) = 3(0.95)^x - 5 \)[/tex], let's analyze the possible outputs.
[tex]\[
\text{If } x \to -\infty, (0.95)^x \to \infty \Rightarrow f(x) \to \infty. \\
\text{If } x \to \infty, (0.95)^x \to 0 \Rightarrow f(x) \to -5.
\][/tex]
Hence the range of [tex]\( f(x) \)[/tex] is all values from [tex]\(-5\)[/tex] (approaching, but not including [tex]\(-5\)[/tex]) to [tex]\( \infty \)[/tex].
Thus, the range of the function is [tex]\( (-5, \infty) \)[/tex].
### Conclusion
Given the analysis, we can now match our findings with the provided options:
- The horizontal asymptote is 3. (False)
- The function is decreasing. (True)
- The function is increasing. (False)
- The range is [tex]\( (-5, \infty) \)[/tex]. (True)
- The horizontal asymptote is -5. (True)
- The range is [tex]\( (3, \infty) \)[/tex]. (False)
Therefore, the true statements are:
- The function is decreasing.
- The range is [tex]\( (-5, \infty) \)[/tex].
- The horizontal asymptote is -5.
The indices of these true statements in the given order are:
[tex]\[ \boxed{[1, 3, 4]} \][/tex]
