Understanding Functions If Point (4,5) Is On The Graph

In mathematics, understanding the relationship between points on a graph and the corresponding function is crucial. This article delves into the concept of how a point on a graph relates to the function's equation. Specifically, we will address the question: If the point $(4, 5)$ lies on the graph of a function, which equation must be true? This question tests the fundamental understanding of function notation and graphical representation. We will explore the correct answer and provide a comprehensive explanation to solidify the underlying principles.

Understanding Function Notation

The cornerstone of this problem lies in understanding function notation. A function, typically denoted as $f(x)$, describes a relationship between an input (usually $x$) and an output (denoted as $f(x)$). The notation $f(x) = y$ means that when the input is $x$, the output is $y$. Graphically, this corresponds to the point $(x, y)$ on the graph of the function. To truly grasp function notation, one must understand that the value inside the parentheses, $x$, represents the input to the function, and $f(x)$ represents the output. This distinction is critical for correctly interpreting and applying functional relationships. For instance, if we have a function $f(x) = x^2$, then $f(3)$ means we substitute $x$ with $3$, resulting in $f(3) = 3^2 = 9$. Thus, the point $(3, 9)$ lies on the graph of this function. The graph of a function is essentially a visual representation of all the input-output pairs that satisfy the function's equation. Therefore, if a point $(a, b)$ lies on the graph of $f(x)$, it implies that $f(a) = b$. Conversely, if we know that $f(a) = b$, we can conclude that the point $(a, b)$ is on the graph. Function notation is a compact and efficient way to express the relationship between inputs and outputs, and it is a foundational concept in algebra and calculus. Without a firm grasp of function notation, it becomes challenging to solve problems involving functions, graphs, and their interconnections. In practical terms, understanding function notation allows us to model real-world phenomena mathematically. For example, if we have a function that describes the height of a projectile over time, function notation enables us to determine the height at any given time by simply substituting the time value into the function. This makes function notation an invaluable tool in various fields, including physics, engineering, economics, and computer science.

Analyzing the Given Point (4, 5)

The problem states that the point $(4, 5)$ is on the graph of a function, which we can denote as $f$. This piece of information is critical. In coordinate geometry, a point is represented as $(x, y)$, where $x$ is the horizontal coordinate (also known as the abscissa) and $y$ is the vertical coordinate (also known as the ordinate). When a point lies on the graph of a function, it means that the coordinates of the point satisfy the function's equation. In the case of the point $(4, 5)$, the $x$-coordinate is $4$, and the $y$-coordinate is $5$. Therefore, if this point is on the graph of the function $f$, it implies that when the input to the function is $4$, the output is $5$. This can be directly translated into function notation as $f(4) = 5$. To further illustrate this concept, consider a simple linear function such as $f(x) = x + 1$. If we want to check if a point, say $(2, 3)$, lies on the graph of this function, we substitute $x = 2$ into the function: $f(2) = 2 + 1 = 3$. Since the output is indeed $3$, the point $(2, 3)$ lies on the graph of $f(x) = x + 1$. Similarly, if we take a point not on the graph, such as $(1, 4)$, substituting $x = 1$ gives $f(1) = 1 + 1 = 2$, which is not equal to the $y$-coordinate $4$. Hence, $(1, 4)$ does not lie on the graph. Applying this understanding to the given point $(4, 5)$, we recognize that the input is $4$ and the output is $5$. This directly corresponds to the function notation $f(4) = 5$. This is the key to solving the problem, as it allows us to eliminate incorrect options and identify the equation that must be true. The visual representation of this concept is also important. Imagine plotting the point $(4, 5)$ on a coordinate plane. If a function's graph passes through this point, it means that at $x = 4$, the function's value (or $y$-coordinate) is $5$. This reinforces the idea that $f(4) = 5$.

Evaluating the Options

Now, let's evaluate the given options based on our understanding of function notation and the information that the point $(4, 5)$ lies on the graph of the function. We know that this implies $f(4) = 5$. Let's go through each option:

• A. $f(5) = 4$: This option states that when the input is $5$, the output is $4$. This is the opposite of what the point $(4, 5)$ tells us. The point $(4, 5)$ implies that when the input is $4$, the output is $5$, not the other way around. Therefore, this option is incorrect. To illustrate why this is incorrect, consider a simple example. If $f(x)$ were a linear function, such as $f(x) = x + 1$, we know that the point $(4, 5)$ satisfies this function because $f(4) = 4 + 1 = 5$. However, $f(5) = 5 + 1 = 6$, which is not equal to $4$. This example highlights that simply swapping the input and output values does not maintain the functional relationship.
• B. $f(5, 4) = 9$: This option introduces a two-variable function notation, $f(5, 4)$. However, the problem only mentions a single-variable function, $f$. Furthermore, even if we were dealing with a two-variable function, the information about the point $(4, 5)$ would not directly translate to $f(5, 4) = 9$. The point $(4, 5)$ only provides information about the function's behavior at a specific input value ($4$). It doesn't give us any information about how the function might behave with two inputs. Therefore, this option is incorrect due to the mismatch in function type (single-variable versus two-variable) and the lack of relevant information from the given point.
• C. $f(4) = 5$: This option perfectly aligns with our understanding. It states that when the input is $4$, the output is $5$, which is exactly what the point $(4, 5)$ tells us. In function notation, this is the correct way to represent that the point $(4, 5)$ lies on the graph of the function $f$. This option accurately reflects the relationship between the input, output, and the function's behavior at the given point. Therefore, this option is the correct one.
• D. $f(5, 4) = 1$: Similar to option B, this option also introduces a two-variable function notation, $f(5, 4)$. Again, the problem is concerned with a single-variable function. The point $(4, 5)$ does not provide any information that would allow us to conclude that $f(5, 4) = 1$. This option is incorrect for the same reasons as option B: mismatch in function type and lack of supporting information from the given point.

Therefore, after carefully evaluating each option, it becomes clear that the only equation that must be true if the point $(4, 5)$ is on the graph of the function is $f(4) = 5$.

Conclusion

In conclusion, understanding the fundamental relationship between a point on a graph and the function's equation is crucial in mathematics. When the point $(4, 5)$ lies on the graph of a function $f$, it directly implies that $f(4) = 5$. This understanding is rooted in the definition of function notation, where $f(x) = y$ represents the output $y$ for an input $x$. By carefully analyzing the given options and applying this principle, we can confidently determine that option C, $f(4) = 5$, is the only equation that must be true. This exercise underscores the importance of mastering function notation and its graphical interpretation, which are essential for solving a wide range of mathematical problems. The ability to translate a point on a graph into function notation, and vice versa, is a foundational skill that extends beyond algebra into calculus and more advanced mathematical disciplines. Furthermore, this problem highlights the significance of paying close attention to the details of the problem statement and avoiding common misconceptions, such as reversing the input and output values or misinterpreting function notation. The correct application of these principles leads to a clear and accurate solution. In essence, this problem serves as a valuable lesson in the power of precise mathematical thinking and the importance of a solid understanding of core concepts.