Ranges Of Exponential Functions F(x) And G(x)

Let's dive into the world of exponential functions and explore how to determine their ranges. Specifically, we'll be looking at two functions: $f(x) = (4/5)^x$ and $g(x) = (4/5)^x + 6$. Understanding the range of a function is crucial as it tells us all the possible output values (y-values) that the function can produce. In this article, we will thoroughly examine these functions, discuss the properties of exponential functions, and step-by-step determine their respective ranges.

Understanding Exponential Functions

To begin, let's understand exponential functions in general. An exponential function has the form $f(x) = a^x$, where 'a' is a positive constant (the base) and 'x' is the variable in the exponent. The behavior of the function depends significantly on the value of 'a'.

When 0 < a < 1, as is the case with our functions where a = 4/5, the function is exponentially decreasing. This means as 'x' increases, the value of $a^x$ decreases, approaching zero but never actually reaching it. Conversely, as 'x' decreases (becomes more negative), the value of $a^x$ increases without bound. This crucial characteristic will help us determine the range of $f(x)$. Another key property is that any positive number raised to any power will always yield a positive result. This fact further constrains the range of $f(x)$. Therefore, our main focus should be on how the base (4/5 in our case) affects the function's behavior as 'x' varies across the entire set of real numbers. By understanding these core principles, we can accurately pinpoint the boundaries of the range.

Moreover, transformations applied to the basic exponential function $a^x$ can shift or stretch the graph, thereby affecting the range. For instance, adding a constant to the function, as seen in $g(x) = (4/5)^x + 6$, shifts the graph vertically. Recognizing these transformations is vital for correctly identifying the range of more complex exponential functions. We need to consider how these shifts influence the possible output values. This involves understanding both the inherent properties of the exponential function and the impact of any additional operations performed on it.

Analyzing the Function f(x) = (4/5)^x

Now, let's analyze the function $f(x) = (4/5)^x$ in detail. As we established, since the base 4/5 is between 0 and 1, this is an exponentially decreasing function. This means that as x gets larger and larger (approaches positive infinity), the value of $(4/5)^x$ gets smaller and smaller, approaching zero. However, it will never actually reach zero because any positive number raised to a power will always be positive. This gives us the lower bound of the range.

On the other hand, as x gets smaller and smaller (approaches negative infinity), the value of $(4/5)^x$ increases without bound, approaching infinity. There is no upper limit to the values that $f(x)$ can take. This behavior is characteristic of decreasing exponential functions. It's like continuously dividing a fraction; the result gets infinitely large as you move towards the negative side of the x-axis. This understanding of asymptotic behavior is essential for determining the range.

Therefore, the range of $f(x)$ includes all positive real numbers. In set notation, we can express this as: $f(x): \{ y \mid y > 0 \}$. This notation concisely conveys that the possible output values (y-values) of $f(x)$ are all real numbers greater than zero. The function approaches zero but never equals it, and it grows without bound as x becomes increasingly negative. This range reflects the fundamental nature of decreasing exponential functions where the output is always positive and approaches zero asymptotically.

Determining the Range of g(x) = (4/5)^x + 6

Next, let's determine the range of the function $g(x) = (4/5)^x + 6$. Notice that this function is simply the function $f(x) = (4/5)^x$ shifted upwards by 6 units. This vertical shift is the key to understanding the range of $g(x)$. The '+ 6' in the function raises every point on the graph of $f(x)$ by 6 units along the y-axis.

We already know that the range of $f(x)$ is all y > 0. This means the lowest value that $(4/5)^x$ can approach is 0 (but it never actually reaches 0). When we add 6 to this, the lowest value that $g(x)$ can approach becomes 0 + 6 = 6. So, $g(x)$ will approach 6 but never actually equal it.

As x approaches negative infinity, $(4/5)^x$ still increases without bound, approaching infinity. Adding 6 to a value that is approaching infinity doesn't change its behavior; the result will still approach infinity. Therefore, there is no upper limit to the values that $g(x)$ can take. The upward shift simply moves the entire range up by 6 units while maintaining the unbounded nature of the function's growth.

Therefore, the range of $g(x)$ is all y > 6. In set notation, this is written as: $g(x): \{ y \mid y > 6 \}$. This indicates that the possible output values of $g(x)$ are all real numbers greater than 6. The function approaches 6 asymptotically but never reaches it, reflecting the upward shift caused by adding 6 to the original function. Understanding this transformation allows us to easily determine the new range.

Conclusion

In conclusion, we have determined the ranges of the functions $f(x) = (4/5)^x$ and $g(x) = (4/5)^x + 6$. The range of $f(x)$ is $y > 0$, meaning it includes all positive real numbers. The range of $g(x)$ is $y > 6$, which is the range of $f(x)$ shifted upwards by 6 units. By understanding the properties of exponential functions and the effects of transformations, we can effectively find the ranges of these functions.

For further reading on exponential functions, you can visit Khan Academy's Exponential Functions.