Finding The X-intercept Of G(x) = Log(x+4) Using F(x) = Log(x)

Introduction

In mathematics, understanding the relationship between functions and their transformations is crucial for solving a variety of problems. This article delves into the concept of finding the x-intercept of a logarithmic function, specifically $g(x) = \log(x+4)$, by leveraging the properties of the parent function $f(x) = \log(x)$. We will explore how transformations affect the graph of a function and how to use this knowledge to determine key features like x-intercepts. The x-intercept is a fundamental characteristic of any function, representing the point where the graph intersects the x-axis. This point is significant because it indicates the value(s) of x for which the function's output, y, is zero. In the context of logarithmic functions, the x-intercept holds particular importance due to the nature of logarithms and their behavior around specific values. Logarithmic functions are the inverse of exponential functions, and their graphs exhibit unique properties, such as a vertical asymptote and a gradual increase or decrease as x varies. Understanding these properties is essential for accurately determining the x-intercept and interpreting its significance within the function's domain and range. The process of finding the x-intercept involves setting the function equal to zero and solving for x. This often requires a solid understanding of logarithmic identities and algebraic manipulation techniques. However, by relating the function to its parent function and considering any transformations applied, we can simplify the process and gain a deeper insight into the function's behavior. In this article, we will walk through the steps of finding the x-intercept of $g(x) = \log(x+4)$, highlighting the connection to the parent function $f(x) = \log(x)$ and explaining the reasoning behind each step. By mastering this concept, readers will be better equipped to analyze and solve problems involving logarithmic functions and their transformations.

Understanding the Parent Function: f(x) = log(x)

To effectively find the x-intercept of $g(x)$, it is essential to first grasp the characteristics of the parent function, $f(x) = \log(x)$. This function serves as the foundation for understanding the transformations applied to obtain $g(x)$. The logarithmic function $f(x) = \log(x)$ is defined as the inverse of the exponential function $y = 10^x$. In simpler terms, the logarithm of a number x is the exponent to which another fixed value, the base, must be raised to produce that number x. In this case, since the base is not explicitly stated, we assume it to be 10, which means we are dealing with the common logarithm. The graph of $f(x) = \log(x)$ has a characteristic shape. It starts from negative infinity as x approaches 0 from the right (i.e., x > 0) and gradually increases as x increases. The graph never touches or crosses the y-axis, indicating a vertical asymptote at x = 0. This asymptote is a crucial feature of logarithmic functions, as it defines the boundary of the function's domain. The x-intercept of $f(x) = \log(x)$ is the point where the graph intersects the x-axis. This occurs when the function's value, y, is equal to 0. To find the x-intercept, we set $f(x) = 0$ and solve for x: $\log(x) = 0$. To solve this equation, we can rewrite it in exponential form. Since the base of the logarithm is 10, we have: $10^0 = x$. Any number raised to the power of 0 is 1, so: $x = 1$. Therefore, the x-intercept of $f(x) = \log(x)$ is (1, 0). This point is a key reference when considering transformations of the function. Understanding the x-intercept of the parent function allows us to predict how the x-intercept of a transformed function will shift based on the type of transformation applied. For instance, horizontal shifts, which we will discuss in the next section, directly affect the x-intercept. Knowing the basic shape and key features of $f(x) = \log(x)$, including its x-intercept and vertical asymptote, is fundamental for analyzing and solving problems involving logarithmic functions. It provides a solid framework for understanding more complex transformations and their impact on the function's graph and properties.

Transformation: Horizontal Shift

The function $g(x) = \log(x+4)$ is a transformation of the parent function $f(x) = \log(x)$. Specifically, it represents a horizontal shift. Understanding transformations is essential in mathematics as it allows us to predict how the graph of a function changes when we modify its equation. In the case of logarithmic functions, transformations can significantly alter the position and shape of the graph, including key features such as the x-intercept and vertical asymptote. A horizontal shift occurs when we add or subtract a constant from the input variable x inside the function. In general, if we have a function $f(x)$ and we replace x with $(x + c)$, where c is a constant, the graph of the function will shift horizontally. If c is positive, the graph shifts to the left by |c| units. If c is negative, the graph shifts to the right by |c| units. In our case, we have $g(x) = \log(x+4)$. Comparing this to the parent function $f(x) = \log(x)$, we can see that x has been replaced with $(x+4)$. This means that the graph of $f(x)$ has been shifted 4 units to the left to obtain the graph of $g(x)$. This horizontal shift directly affects the vertical asymptote and the x-intercept of the function. The vertical asymptote of $f(x) = \log(x)$ is x = 0. When the graph is shifted 4 units to the left, the vertical asymptote also shifts 4 units to the left, becoming x = -4 for the function $g(x) = \log(x+4)$. This shift in the vertical asymptote is a crucial aspect of the transformation, as it defines the new domain of the function. The domain of $f(x) = \log(x)$ is all x > 0, while the domain of $g(x) = \log(x+4)$ is all x > -4. The x-intercept, as mentioned earlier, is the point where the graph intersects the x-axis. For $f(x) = \log(x)$, the x-intercept is (1, 0). The horizontal shift will also move this point. Since the graph is shifted 4 units to the left, the x-intercept will also shift 4 units to the left. To find the new x-intercept, we need to determine the x-coordinate of the point where $g(x) = 0$. This can be done by setting the function equal to zero and solving for x, as we will demonstrate in the next section. Understanding the effect of horizontal shifts on logarithmic functions is fundamental for analyzing and graphing these functions. By recognizing the transformation, we can quickly determine the new vertical asymptote and make informed predictions about the x-intercept. This knowledge simplifies the process of solving problems involving logarithmic functions and their transformations.

Finding the x-intercept of g(x) = log(x+4)

Now that we understand the transformation applied to the parent function, we can find the x-intercept of $g(x) = \log(x+4)$. The x-intercept is the point where the graph of the function intersects the x-axis. At this point, the y-coordinate is zero. Therefore, to find the x-intercept, we need to solve the equation $g(x) = 0$. This means setting the function equal to zero and solving for x: $\log(x+4) = 0$. To solve this logarithmic equation, we can rewrite it in exponential form. Remember that the common logarithm (log) has a base of 10. So, the equation $\log(x+4) = 0$ can be rewritten as: $10^0 = x + 4$. Any non-zero number raised to the power of 0 is 1, so we have: $1 = x + 4$. Now, we can solve for x by subtracting 4 from both sides of the equation: $1 - 4 = x$, which simplifies to: $x = -3$. Therefore, the x-coordinate of the x-intercept is -3. The x-intercept is the point (-3, 0). This result aligns with our understanding of the horizontal shift. We know that the x-intercept of the parent function $f(x) = \log(x)$ is (1, 0). The function $g(x) = \log(x+4)$ is a horizontal shift of $f(x)$ by 4 units to the left. So, we would expect the x-intercept to also shift 4 units to the left. Starting from x = 1, shifting 4 units to the left gives us x = 1 - 4 = -3, which confirms our calculated x-intercept of (-3, 0). It's important to note that we should always check if the x-intercept we found is within the domain of the function. The domain of $g(x) = \log(x+4)$ is all x such that $x + 4 > 0$, which means x > -4. Our x-intercept, x = -3, satisfies this condition, so it is a valid solution. In summary, to find the x-intercept of a logarithmic function, we set the function equal to zero, rewrite the equation in exponential form, and solve for x. We can also use our understanding of transformations to predict the x-intercept and verify our result. This method provides a systematic approach to finding x-intercepts and enhances our understanding of logarithmic functions and their properties.

Conclusion

In conclusion, finding the x-intercept of $g(x) = \log(x+4)$ involves understanding the parent function $f(x) = \log(x)$, recognizing the horizontal shift transformation, and solving the equation $g(x) = 0$. By understanding the fundamental characteristics of logarithmic functions and how transformations affect their graphs, we can efficiently determine key features such as the x-intercept. The x-intercept, which is the point where the graph intersects the x-axis, is found by setting the function equal to zero and solving for x. In this case, we set $\log(x+4) = 0$, which led us to the solution $x = -3$. Therefore, the x-intercept of $g(x) = \log(x+4)$ is (-3, 0). The horizontal shift of 4 units to the left, from $f(x)$ to $g(x)$, plays a crucial role in determining the new x-intercept. The x-intercept of the parent function, (1, 0), is shifted by the same amount, leading us to expect the x-intercept of $g(x)$ to be at x = -3. This understanding of transformations simplifies the process of finding the x-intercept and provides a way to verify our calculations. Throughout this article, we have emphasized the importance of relating transformed functions to their parent functions. This approach not only helps in finding x-intercepts but also provides a deeper understanding of how transformations affect other features of the graph, such as the vertical asymptote and the overall shape. Understanding the domain of the function is also critical. We verified that our solution, x = -3, is within the domain of $g(x) = \log(x+4)$, which is x > -4. This step ensures that our solution is valid and meaningful. By mastering the concepts discussed in this article, readers will be well-equipped to tackle problems involving logarithmic functions and their transformations. The ability to find x-intercepts, understand transformations, and relate functions to their parent functions are essential skills in mathematics, particularly in calculus and other advanced topics. This systematic approach provides a solid foundation for further exploration of mathematical concepts and problem-solving strategies.