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

Let's delve into the world of exponential functions by examining three specific examples: $f(x)=\frac{3}{2}(4)^x$, $g(x)=\frac{3}{2}(4)^{-x}$, and $h(x)=-\frac{3}{2}(4)^x$. Understanding the characteristics of these functions, particularly their domains and ranges, is crucial in mathematics. The domain of a function encompasses all possible input values (x-values) for which the function is defined, while the range represents all possible output values (y-values) that the function can produce. In this comprehensive analysis, we will thoroughly explore the domains and ranges of these three functions, providing clear explanations and justifications for each case.

When we consider the function f(x) = (3/2)(4)^x, we are dealing with a standard exponential function. In this context, the variable 'x' appears as the exponent. A critical property of exponential functions with a positive base (in this case, 4) is that they are defined for all real numbers. This means that we can input any real number value for 'x', and the function will produce a valid output. There are no restrictions on the values that 'x' can take. Therefore, the domain of f(x) is the set of all real numbers, often denoted as (-∞, ∞). To make it more clear, you can choose any real number, positive, negative, or zero, plug it in for x, and you'll always get a real number result. There are no square roots of negative numbers or division by zero situations that would cause a problem.

Now, let's shift our focus to the range of f(x). Since the base 4 is a positive number greater than 1, the exponential term (4)^x will always be a positive value for any real number x. When we multiply this positive value by the constant 3/2, the result remains positive. As x approaches negative infinity, (4)^x approaches 0, so (3/2)(4)^x also approaches 0, but it never actually reaches 0. As x approaches positive infinity, (4)^x grows without bound, and so does (3/2)(4)^x. Therefore, the range of f(x) consists of all positive real numbers, which can be written as (0, ∞). To be more specific, f(x) will always output a number greater than zero, but it can get infinitely large. There's no upper bound on the output, but zero acts as a lower bound that the function never touches.

Turning our attention to the function g(x) = (3/2)(4)^-x, we again encounter an exponential function, but with a crucial difference: the exponent is -x. This negative sign in the exponent introduces a reflection across the y-axis compared to the function f(x). However, it does not affect the domain. Just like f(x), g(x) is defined for all real numbers. We can substitute any real number for x, and the function will produce a valid output. The negative sign in the exponent simply means that we are dealing with the reciprocal of 4 raised to the power of x. This doesn't introduce any domain restrictions, like division by zero or taking the square root of a negative number. Therefore, the domain of g(x) is also the set of all real numbers, (-∞, ∞). Essentially, g(x) is the same as (3/2)*(1/4)^x, which is still an exponential function defined for all real numbers.

When we analyze the range of g(x), we observe similar behavior to f(x), but with a reversed trend. The term (4)^-x is equivalent to 1/(4^x). As x approaches positive infinity, (4)^-x approaches 0, and as x approaches negative infinity, (4)^-x grows without bound. Multiplying by the positive constant 3/2 maintains the positivity of the expression. Thus, g(x) will always yield positive outputs, approaching 0 but never reaching it, and increasing without bound as x becomes increasingly negative. Consequently, the range of g(x) is the set of all positive real numbers, represented as (0, ∞). While the graph of g(x) will be a decreasing exponential function, as opposed to the increasing f(x), the range remains the same because the base is still positive, and the constant multiplier is also positive.

Lastly, we investigate the function h(x) = -(3/2)(4)^x. This function is closely related to f(x), with the key difference being the negative sign in front of the expression. This negative sign introduces a vertical reflection across the x-axis. Much like f(x), the domain of h(x) encompasses all real numbers. The base of the exponential term is still positive, and there are no restrictions on the values that x can take. Therefore, the domain of h(x) is (-∞, ∞). You can plug in any real number for x, and you will get a real number output. There are no situations where the function is undefined.

The range of h(x), however, is significantly different from f(x) and g(x) due to the negative sign. The term (4)^x will always be positive, but when multiplied by -3/2, the result becomes negative. As x approaches negative infinity, (4)^x approaches 0, and so does -(3/2)(4)^x, but it remains negative. As x approaches positive infinity, (4)^x grows without bound, and -(3/2)(4)^x decreases without bound, becoming increasingly negative. Therefore, the range of h(x) consists of all negative real numbers. This can be written as (-∞, 0). The function will output values that are always less than zero, approaching zero but never reaching it, and decreasing infinitely as x becomes larger. The negative sign flips the range of f(x) from positive to negative.

To consolidate our findings, let's present a summary table of the domains and ranges for each function:

This table succinctly illustrates the key differences and similarities in the domains and ranges of the three functions.

In conclusion, our exploration of the exponential functions f(x), g(x), and h(x) has revealed critical insights into their domains and ranges. We've established that exponential functions of the form a^x, where a is a positive constant, are defined for all real numbers, leading to a domain of (-∞, ∞). However, the range is significantly influenced by the base and any constant multipliers. In our case, the functions f(x) and g(x) both have a range of (0, ∞), indicating that their outputs are always positive. This is due to the positive base and the positive constant multiplier (3/2). The function h(x), on the other hand, features a negative constant multiplier, resulting in a range of (-∞, 0), signifying that its outputs are always negative. Understanding these nuances is crucial for effectively working with exponential functions in various mathematical contexts. Through a deeper understanding of these functions, one can confidently apply them in diverse scenarios, reinforcing their mathematical knowledge and analytical skills.

The analysis of domains and ranges of exponential functions is a fundamental concept in mathematics. The domain of the functions $f(x)$, $g(x)$, and $h(x)$ are all real numbers, since there are no restrictions on the values of $x$ that can be input into these functions. The range of $f(x)$ and $g(x)$ is $(0, ext{infinity})$, since the exponential term is always positive and the constant factor is also positive. The range of $h(x)$ is $(- ext{infinity}, 0)$, since the exponential term is always positive but the constant factor is negative.

Here's a breakdown of why the statements about the domain and range of the functions are true or false:

• f(x) has a domain of all real numbers. - This statement is TRUE. As explained above, exponential functions with a constant base are defined for all real numbers.

In summary, analyzing these exponential functions by pinpointing their domains and ranges offers valuable insights. The domain, representing valid inputs, remains consistent for all three functions, encompassing all real numbers. Conversely, the range, denoting feasible outputs, demonstrates more variability, influenced by coefficients and reflections. This analysis reinforces fundamental mathematical concepts and underscores the significance of thorough function dissection in problem-solving endeavors.