Does (1/2, √5) Lie On The Graph Of F(x) = 5^x?

Let's dive into the question of whether the point (1/2, √5) actually sits on the graph of the function f(x) = 5^x. This involves a bit of function evaluation and understanding how graphs work. We'll break it down step by step to make sure it's crystal clear. Understanding how points relate to functions is crucial in mathematics, especially in areas like calculus and analysis. The point (1/2, √5) represents a specific location on the coordinate plane, and the function f(x) = 5^x defines a curve. For the point to lie on the graph, its coordinates must satisfy the function's equation. This means that when we plug in the x-coordinate (1/2) into the function, we should get the y-coordinate (√5). If this condition is met, then we can confidently say that the point lies on the graph.

Understanding the Function f(x) = 5^x

Before we can test our point, let's quickly revisit what the function f(x) = 5^x means. This is an exponential function, where the base is 5 and the exponent is x. Exponential functions are characterized by rapid growth, and they play a significant role in modeling various real-world phenomena, such as population growth, compound interest, and radioactive decay. The base of the exponent, in this case 5, determines the rate of growth. As x increases, the value of f(x) increases exponentially. To evaluate f(x) for a specific value of x, we simply substitute that value into the function. For example, f(2) = 5^2 = 25. This means that the point (2, 25) lies on the graph of f(x) = 5^x. Similarly, f(0) = 5^0 = 1, so the point (0, 1) also lies on the graph. Understanding these basic concepts is essential for tackling more complex problems involving exponential functions.

Exponential functions have some key properties that are important to keep in mind. They are always positive, meaning that f(x) will always be greater than 0, regardless of the value of x. They also have a horizontal asymptote at y = 0, which means that the graph approaches the x-axis as x becomes very negative but never actually touches it. The domain of an exponential function is all real numbers, but the range is only positive real numbers. These properties help us understand the overall behavior of the function and its graph. Furthermore, exponential functions are closely related to logarithmic functions, which are their inverses. The relationship between exponential and logarithmic functions is a fundamental concept in mathematics and is used extensively in various applications.

Evaluating f(1/2)

Now, let's plug in the x-coordinate of our point, which is 1/2, into the function f(x) = 5^x. This gives us f(1/2) = 5^(1/2). But what does 5 raised to the power of 1/2 actually mean? Remember, a fractional exponent indicates a root. Specifically, x^(1/n) is the same as the nth root of x. In our case, 5^(1/2) is the same as the square root of 5, which is written as √5. Understanding fractional exponents is crucial for working with exponential functions and radicals. They provide a concise way to express roots and simplify calculations. For example, 8^(1/3) is the cube root of 8, which is 2. Similarly, 16^(1/4) is the fourth root of 16, which is 2. These concepts are used extensively in algebra and calculus.

So, we've found that f(1/2) = √5. This means that when x is 1/2, the function's value, f(x), is √5. This is a critical step in determining whether the point (1/2, √5) lies on the graph. By evaluating the function at the x-coordinate of the point, we can directly compare the result with the y-coordinate. If the result matches the y-coordinate, then the point lies on the graph. If not, then the point does not lie on the graph. In our case, the y-coordinate of the point is also √5, which is the same as the value we obtained for f(1/2). This confirms that the point (1/2, √5) satisfies the function's equation and therefore lies on the graph.

Does the Point Lie on the Graph?

We've done the math! We found that f(1/2) = √5. This perfectly matches the y-coordinate of the point (1/2, √5). Therefore, we can confidently conclude that the point (1/2, √5) does lie on the graph of the function f(x) = 5^x. This is because the x and y coordinates of the point satisfy the equation y = 5^x. When we substitute x = 1/2 into the equation, we get y = 5^(1/2) = √5, which is the y-coordinate of the point. This confirms that the point lies on the graph. Visualizing the graph can also help to reinforce this understanding. If we were to plot the graph of f(x) = 5^x, we would see that the point (1/2, √5) falls directly on the curve.

This exercise highlights the fundamental relationship between functions and their graphs. A point lies on the graph of a function if and only if its coordinates satisfy the function's equation. This principle is used extensively in mathematics to analyze functions and their properties. For example, it is used to find the points where a function intersects the axes, to determine the intervals where a function is increasing or decreasing, and to identify local maxima and minima. Understanding this relationship is essential for solving a wide range of problems in calculus and other areas of mathematics.

Implications and Further Exploration

Knowing that (1/2, √5) lies on the graph of f(x) = 5^x opens doors to further exploration. We could, for instance, consider the inverse function of f(x) = 5^x, which is a logarithmic function. The point (√5, 1/2) would then lie on the graph of the inverse function. Exploring the properties of inverse functions can provide valuable insights into the behavior of the original function. Additionally, we can use this information to solve equations involving exponential functions. For example, if we want to find the value of x for which 5^x = √5, we can directly conclude that x = 1/2 based on our previous calculations. This demonstrates the practical applications of understanding the relationship between points and graphs of functions.

Furthermore, we can extend this concept to other types of functions, such as polynomial, trigonometric, and rational functions. For each type of function, the principle remains the same: a point lies on the graph if its coordinates satisfy the function's equation. By applying this principle, we can analyze and understand the behavior of various functions and their graphs. This is a fundamental skill in mathematics and is used extensively in various fields, including physics, engineering, and economics.

In conclusion, we've confirmed that the point (1/2, √5) does indeed lie on the graph of the function f(x) = 5^x by evaluating the function at x = 1/2 and verifying that the result matches the y-coordinate of the point. This exercise illustrates a fundamental concept in mathematics – the relationship between a function and its graph. For further exploration of exponential functions, you might find resources on websites like Khan Academy helpful.