Domain And Range Of F(x) = Log(x-1) + 2 Explained

In the realm of mathematics, understanding the domain and range of a function is paramount to grasping its behavior and characteristics. These two fundamental concepts define the set of possible input values (domain) and the resulting output values (range) of a function. Let's embark on an in-depth exploration of the logarithmic function f(x) = log(x-1) + 2, dissecting its domain and range to gain a comprehensive understanding.

Demystifying the Domain of f(x) = log(x-1) + 2

The domain, in essence, represents the set of all permissible input values (x-values) for which the function yields a defined output. For logarithmic functions, a critical constraint arises from the nature of logarithms themselves. Logarithms are only defined for positive arguments. In simpler terms, you can only take the logarithm of a positive number. This restriction stems from the very definition of a logarithm as the inverse operation of exponentiation. Consider the expression logₐ(b) = c. This translates to aᶜ = b. If 'b' (the argument of the logarithm) is negative or zero, there's no real number 'c' that can satisfy this equation for a positive base 'a'. This is because any positive number raised to any power will always result in a positive number. Therefore, the argument of the logarithm, (x-1) in our case, must be strictly greater than zero.

Applying this principle to our function, f(x) = log(x-1) + 2, we need to ensure that the argument of the logarithm, (x-1), is positive. Mathematically, this translates to the inequality:

x - 1 > 0

To solve this inequality, we simply add 1 to both sides:

x > 1

This inequality reveals that the domain of the function f(x) = log(x-1) + 2 consists of all real numbers x that are strictly greater than 1. In interval notation, we express this domain as (1, ∞). This means that the function is defined for any value of x greater than 1, but it is undefined for x values less than or equal to 1. The vertical asymptote of the function lies at x = 1, which means the function approaches this line but never actually touches or crosses it. This is a crucial characteristic of logarithmic functions, directly stemming from the requirement of a positive argument for the logarithm. Understanding this limitation is key to accurately interpreting the behavior and graph of logarithmic functions.

Unveiling the Range of f(x) = log(x-1) + 2

The range of a function encompasses the set of all possible output values (y-values) that the function can produce. To determine the range of f(x) = log(x-1) + 2, we need to consider how the logarithmic component and the constant term influence the output. The base logarithmic function, log(x), without any transformations, has a range of all real numbers. This means that as x varies over its domain (positive real numbers), the logarithm can take on any real value, both positive and negative. The logarithmic function grows very slowly as x increases, but it does increase without bound. Similarly, as x approaches 0 from the positive side, the logarithm decreases without bound, approaching negative infinity.

In our function, f(x) = log(x-1) + 2, the logarithmic part, log(x-1), still retains this fundamental characteristic of having a range of all real numbers. The horizontal shift caused by the (x-1) term inside the logarithm only affects the domain, as we discussed earlier, but it does not alter the range. The vertical shift caused by the +2 term, however, plays a crucial role in understanding the range. The +2 shifts the entire graph of the function upwards by 2 units. However, since the logarithmic component can take on any real value, adding 2 to it doesn't restrict the possible output values. The function can still attain any real value, just shifted upwards by 2 units.

Therefore, the range of f(x) = log(x-1) + 2 is also all real numbers. This can be expressed in interval notation as (-∞, ∞). The function spans the entire vertical axis, meaning there is no upper or lower bound on the possible output values. The +2 simply shifts the entire range upwards without compressing or limiting it. This understanding of the range, coupled with our earlier analysis of the domain, provides a complete picture of the possible input-output relationships for this logarithmic function.

Synthesizing the Domain and Range

Having meticulously examined both the domain and range of f(x) = log(x-1) + 2, we can now consolidate our findings. The domain of the function is x > 1, or in interval notation, (1, ∞). This restriction arises from the fundamental requirement that the argument of a logarithm must be positive. The range of the function, on the other hand, encompasses all real numbers, represented as (-∞, ∞). This stems from the inherent nature of the logarithmic function, which can produce any real number as output, coupled with the vertical shift of +2, which doesn't limit the range. By combining our understanding of the domain and range, we gain a powerful insight into the overall behavior of the function. We know that the function is only defined for x values greater than 1, and for these x values, the function can take on any real number as its output. This comprehensive understanding forms the foundation for further analysis, such as graphing the function, finding its inverse, or solving related equations and inequalities.

Visualizing the Domain and Range on a Graph

Graphing the function f(x) = log(x-1) + 2 provides a powerful visual representation of its domain and range. When you plot the graph, you'll observe that it exists only to the right of the vertical line x = 1. This visually confirms our earlier determination of the domain as (1, ∞). The graph approaches the line x = 1 asymptotically, getting infinitely close but never actually touching it. This vertical asymptote is a direct consequence of the domain restriction.

Furthermore, you'll notice that the graph extends infinitely upwards and downwards, covering the entire vertical axis. This visually demonstrates the range of the function as (-∞, ∞). There are no horizontal boundaries to the graph; it continues to rise and fall indefinitely. The graph starts at very large negative values as it approaches the vertical asymptote and slowly increases as x increases. The +2 in the function shifts the entire graph upwards by 2 units compared to the basic log(x-1) function, but this vertical shift doesn't change the overall range.

The graph serves as a valuable tool for reinforcing the concepts of domain and range. It provides a concrete visual representation of the set of permissible input values and the resulting output values. By examining the graph, we can readily identify the domain as the set of x-values for which the graph exists, and the range as the set of y-values that the graph covers. This visual confirmation complements our analytical determination of the domain and range, solidifying our understanding of the function's behavior.

Practical Implications of Domain and Range

The concepts of domain and range are not merely abstract mathematical ideas; they have significant practical implications in various fields. Consider, for instance, scenarios involving logarithmic scales, such as the Richter scale for measuring earthquake magnitude or the pH scale for measuring acidity. Logarithmic scales are used to represent large ranges of values in a compressed manner. In these contexts, understanding the domain and range of the logarithmic function is crucial for correctly interpreting the scale and making meaningful comparisons.

For example, the Richter scale uses a base-10 logarithm to quantify earthquake magnitude. The magnitude is related to the amplitude of seismic waves recorded by seismographs. Since the logarithm is only defined for positive values, the amplitude of the seismic waves must be a positive quantity. This represents the domain restriction. The range of the Richter scale is theoretically unbounded, but in practice, there are physical limits to the magnitude of earthquakes. Understanding this range helps in assessing the potential severity of seismic events.

Similarly, in chemistry, the pH scale uses a negative base-10 logarithm to measure the concentration of hydrogen ions in a solution. The pH value indicates the acidity or alkalinity of the solution. The concentration of hydrogen ions must be positive, again reflecting the domain constraint of the logarithm. The pH scale typically ranges from 0 to 14, representing a vast range of hydrogen ion concentrations. Knowing the range of the pH scale is essential for understanding the chemical properties of solutions and their interactions.

These examples illustrate the practical relevance of domain and range in real-world applications. By understanding these concepts, we can effectively model and interpret phenomena that involve logarithmic relationships, making informed decisions and drawing accurate conclusions.

Conclusion: Mastering Domain and Range for Logarithmic Functions

In conclusion, deciphering the domain and range of logarithmic functions, such as f(x) = log(x-1) + 2, is a cornerstone of mathematical understanding. The domain, restricted by the necessity of a positive argument for the logarithm, dictates the permissible input values. In this case, the domain is x > 1, a crucial constraint that shapes the function's behavior. The range, encompassing all real numbers, reveals the function's capacity to produce any output value, a characteristic inherent to logarithmic functions. Visualizing the function's graph reinforces these concepts, providing a concrete representation of the domain and range.

Moreover, the practical implications of domain and range extend beyond theoretical mathematics. Logarithmic scales, prevalent in fields like seismology and chemistry, underscore the real-world significance of these concepts. By mastering the domain and range of logarithmic functions, we equip ourselves with the tools to analyze, interpret, and apply these functions in diverse contexts. This understanding not only enhances our mathematical prowess but also empowers us to make informed decisions in various scientific and practical domains.