Find (f-g)(x) If F(x)=4^x-8 And G(x)=5x+6
Let's dive into a fun math problem where we need to find the difference between two functions. We're given two functions: f(x) which equals 4^x - 8, and g(x) which equals 5x + 6. Our mission, should we choose to accept it, is to find what (f-g)(x) is. This simply means we need to subtract the function g(x) from the function f(x). Ready? Let's get started!
Understanding the Functions
Before we jump into the subtraction, let's make sure we fully grasp what each function represents. The function f(x) = 4^x - 8 is an exponential function with a constant subtracted from it. Exponential functions grow very rapidly as x increases, especially when the base (in this case, 4) is greater than 1. The "- 8" simply shifts the entire graph down by 8 units on the y-axis. So, for example, when x = 0, f(0) = 4^0 - 8 = 1 - 8 = -7. When x = 1, f(1) = 4^1 - 8 = 4 - 8 = -4. When x = 2, f(2) = 4^2 - 8 = 16 - 8 = 8. You can see how quickly the value of f(x) increases.
On the other hand, the function g(x) = 5x + 6 is a linear function. Linear functions have a constant rate of change, meaning for every increase of 1 in x, g(x) increases by a constant amount (in this case, 5). The "+ 6" is the y-intercept, meaning the line crosses the y-axis at the point (0, 6). For example, when x = 0, g(0) = 5(0) + 6 = 6. When x = 1, g(1) = 5(1) + 6 = 11. When x = 2, g(2) = 5(2) + 6 = 16. The value of g(x) increases linearly.
Understanding the behavior of these two functions will help us appreciate how their difference, (f-g)(x), behaves as well.
Performing the Subtraction: (f-g)(x)
Now for the main event: finding (f-g)(x). Remember, (f-g)(x) simply means f(x) - g(x). So, we substitute the expressions for f(x) and g(x):
(f-g)(x) = (4^x - 8) - (5x + 6)
To simplify this, we need to distribute the negative sign to both terms inside the second parenthesis:
(f-g)(x) = 4^x - 8 - 5x - 6
Now, we combine the constant terms (-8 and -6):
(f-g)(x) = 4^x - 5x - 14
And that's it! We've found (f-g)(x). It's equal to 4^x - 5x - 14. This resulting function is neither purely exponential nor purely linear; it's a combination of both.
Putting it all together
So, to recap, we were given two functions, f(x) = 4^x - 8 and g(x) = 5x + 6, and asked to find (f-g)(x). We understood that (f-g)(x) means f(x) - g(x). We substituted the expressions for f(x) and g(x), distributed the negative sign, and combined like terms. This gave us our final answer:
(f-g)(x) = 4^x - 5x - 14
This function represents the difference between the exponential function f(x) and the linear function g(x). As x increases, the exponential term 4^x will eventually dominate, causing (f-g)(x) to increase rapidly.
Let's consider a few examples to solidify our understanding. When x = 0:
(f-g)(0) = 4^0 - 5(0) - 14 = 1 - 0 - 14 = -13
When x = 1:
(f-g)(1) = 4^1 - 5(1) - 14 = 4 - 5 - 14 = -15
When x = 2:
(f-g)(2) = 4^2 - 5(2) - 14 = 16 - 10 - 14 = -8
When x = 3:
(f-g)(3) = 4^3 - 5(3) - 14 = 64 - 15 - 14 = 35
Notice how the value starts increasing as x increases. This is due to the exponential term 4^x becoming significantly larger.
Visualizing the Functions
To further enhance our understanding, it can be helpful to visualize these functions graphically. Imagine the graph of f(x) = 4^x - 8. It starts below the x-axis and rises rapidly as x increases. Now imagine the graph of g(x) = 5x + 6, a straight line with a positive slope. The graph of (f-g)(x) represents the vertical distance between these two graphs at each value of x. Initially, g(x) is greater than f(x), so (f-g)(x) is negative. However, as x increases, f(x) eventually becomes greater than g(x), and (f-g)(x) becomes positive and grows rapidly.
Conclusion
In conclusion, we successfully found (f-g)(x) given f(x) = 4^x - 8 and g(x) = 5x + 6. The process involved understanding function notation, substituting the given expressions, distributing the negative sign, and combining like terms. The final result, (f-g)(x) = 4^x - 5x - 14, is a new function that represents the difference between the original two functions. By understanding the behavior of exponential and linear functions, we gained a deeper insight into how (f-g)(x) behaves as well. Keep practicing with different functions and operations to strengthen your understanding of function manipulation!
For more information on functions and their operations, you can visit Khan Academy's Functions page.
