Is F(x) = 3^x - 2^x - 2 Monotone A Deep Dive Into Function Behavior
Hey guys! Today, we're diving into a fascinating question about the function f(x) = 3^x - 2^x - 2. Specifically, we want to figure out if this function is monotone. Now, what does monotone even mean? Well, in simple terms, a function is monotone if it's either always increasing or always decreasing. Think of it like a hill – you're either always climbing up or always sliding down. There's no going up and down.
So, let's get into it. Our function is f(x) = 3^x - 2^x - 2. The question we're tackling is: Is this function monotone? It seems straightforward at first glance, but there are some interesting twists and turns, especially when we consider different ranges of x.
Initial Observations
When we look at x > 0, things seem pretty clear. As x gets bigger, 3^x grows much faster than 2^x. This makes intuitive sense, right? A larger base raised to a power will grow more rapidly. So, 3^x will eventually dwarf 2^x, and the function f(x) will increase without bound. We can say that as x approaches infinity, f(x) also approaches infinity. This strongly suggests that f(x) is increasing for positive x values.
But hold on! Things get a bit trickier when we consider x < 0. This is where the “weird thing happens,” as someone pointed out. We need to dig deeper into what's going on with negative values of x. When dealing with negative exponents, we're essentially talking about reciprocals. For instance, 3^-x is the same as 1/(3^x). So, as x becomes a large negative number, 3^x and 2^x both approach zero, but at different rates. This difference in rates is where the complexity arises.
The Curious Case of Negative x
Let’s think about the limit of f(x) as x approaches negative infinity. As x becomes increasingly negative, both 3^x and 2^x approach 0. However, the function has that “- 2” term hanging around. So, what happens to f(x)?
lim (x→ -∞) f(x) = lim (x→ -∞) (3^x - 2^x - 2) = 0 - 0 - 2 = -2
This means that as x goes towards negative infinity, f(x) approaches -2. This is a crucial piece of the puzzle. It tells us that the function doesn't decrease indefinitely; it has a lower bound. But does this mean it’s monotone for negative x? Not necessarily. We need to understand how the function behaves as it approaches this limit.
Diving Deeper: Analyzing the Derivative
To get a better handle on the function's behavior, especially for negative x, we can turn to calculus. Specifically, let's look at the derivative of f(x). The derivative, f’(x), will tell us the rate of change of the function. If f’(x) is always positive, the function is increasing. If it's always negative, the function is decreasing. And if it changes sign, well, then we know the function isn't monotone.
So, let's calculate the derivative:
f(x) = 3^x - 2^x - 2 f’(x) = (3^x) * ln(3) - (2^x) * ln(2)
Now, we need to analyze the sign of f’(x). This is where things get interesting. We want to know if (3^x) * ln(3) - (2^x) * ln(2) is always positive, always negative, or if it changes sign.
Analyzing the Sign of the Derivative
Let's rewrite the derivative inequality:
(3^x) * ln(3) - (2^x) * ln(2) > 0 (3^x) * ln(3) > (2^x) * ln(2) (3^x) / (2^x) > ln(2) / ln(3) (3/2)^x > ln(2) / ln(3)
Now, let's take the natural logarithm of both sides:
x * ln(3/2) > ln(ln(2) / ln(3))
Since ln(3/2) is positive, we can divide both sides without flipping the inequality:
x > ln(ln(2) / ln(3)) / ln(3/2)
Let's call C = ln(ln(2) / ln(3)) / ln(3/2). This is just a constant. Using a calculator, we find that C is approximately -2.44.
So, f’(x) > 0 when x > C (approximately -2.44), and f’(x) < 0 when x < C. This is a crucial finding! It tells us that the derivative changes sign. The function is decreasing for x < C and increasing for x > C.
Conclusion: Is f(x) Monotone?
Drumroll, please! The answer is no, the function f(x) = 3^x - 2^x - 2 is not monotone over its entire domain. We've shown that it decreases for x < C (where C ≈ -2.44) and increases for x > C. This means there's a point where the function changes direction, shattering any hope of it being monotone.
However, we can say something more specific. The function is monotone on the intervals (-∞, C] and [C, ∞). On each of these intervals, the function is either strictly decreasing or strictly increasing. So, it's piecewise monotone, if you will.
Key Takeaways
- Monotonicity means a function is either always increasing or always decreasing.
- The function f(x) = 3^x - 2^x - 2 is increasing for x > 0.
- The limit of f(x) as x approaches negative infinity is -2.
- Analyzing the derivative f’(x) helps determine where a function is increasing or decreasing.
- f(x) is decreasing for x < C and increasing for x > C, where C ≈ -2.44.
- Therefore, f(x) is not monotone over its entire domain but is monotone on specific intervals.