One-to-One Function And Inverse Function Calculation For F(x) = 7/(x+9)
Determining whether a function is one-to-one and finding its inverse is a fundamental concept in mathematics. In this article, we will explore the process of identifying one-to-one functions and deriving the formula for their inverses, focusing on the example function f(x) = 7/(x+9). Let's delve into the key concepts and techniques involved.
Understanding One-to-One Functions
One-to-one functions, also known as injective functions, are functions where each element in the range corresponds to exactly one element in the domain. In simpler terms, a function is one-to-one if no two different inputs produce the same output. This unique mapping between inputs and outputs is crucial for the existence of an inverse function. To determine if a function is one-to-one, we can use two primary methods: the horizontal line test and the algebraic approach.
The horizontal line test is a graphical method. If any horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one. This is because the points of intersection represent different x-values (inputs) that yield the same y-value (output), violating the one-to-one condition. For example, consider a parabola. A horizontal line can intersect it at two points, indicating that the function is not one-to-one.
The algebraic approach involves assuming that f(x₁) = f(x₂) for some x₁ and x₂ in the domain and then showing that x₁ must be equal to x₂. If we can demonstrate that the equality of the function values implies the equality of the inputs, then the function is one-to-one. This method provides a rigorous proof and is particularly useful for functions that are not easily graphed. For instance, if we have a function f(x) = 2x + 3, setting f(x₁) = f(x₂) gives us 2x₁ + 3 = 2x₂ + 3. Simplifying this equation, we find that x₁ = x₂, proving that the function is one-to-one.
In the context of our example function, f(x) = 7/(x+9), we can apply the algebraic approach. Assume f(x₁) = f(x₂), which means 7/(x₁+9) = 7/(x₂+9). Cross-multiplying gives us 7(x₂+9) = 7(x₁+9). Dividing both sides by 7, we have x₂+9 = x₁+9. Subtracting 9 from both sides, we obtain x₁ = x₂. This demonstrates that the function f(x) = 7/(x+9) is indeed one-to-one. Understanding these methods is crucial in assessing the invertibility of a function, which is our next step.
Verifying f(x) = 7/(x+9) is One-to-One
To definitively verify that the function f(x) = 7/(x+9) is one-to-one, we will employ the algebraic method, which provides a rigorous proof. The algebraic approach is particularly useful for functions that may not be easily visualized graphically, allowing us to establish a direct relationship between the inputs and outputs.
The core principle of this method involves assuming that two outputs of the function are equal, i.e., f(x₁) = f(x₂), and then demonstrating that this equality implies the inputs must also be equal, i.e., x₁ = x₂. If we can successfully show this implication, it confirms that no two different inputs produce the same output, which is the defining characteristic of a one-to-one function.
Let's apply this to our function. Assume that f(x₁) = f(x₂). This means that:
7/(x₁+9) = 7/(x₂+9)
To solve this equation, we can start by cross-multiplying, which eliminates the fractions and simplifies the expression. This gives us:
7(x₂+9) = 7(x₁+9)
Next, we can divide both sides of the equation by 7, further simplifying it:
x₂+9 = x₁+9
Now, to isolate the variables x₁ and x₂, we subtract 9 from both sides of the equation:
x₂ = x₁
This final result, x₁ = x₂, is the key to our proof. It demonstrates that if f(x₁) = f(x₂), then it must be the case that x₁ = x₂. This confirms that no two different inputs can produce the same output for the function f(x) = 7/(x+9). Therefore, we can confidently conclude that the function is one-to-one.
By verifying that f(x) = 7/(x+9) is a one-to-one function, we establish the necessary condition for the existence of an inverse function. A function must be one-to-one to have an inverse, as the inverse function will reverse the mapping, and each output must map back to a unique input. Now that we have confirmed that our function meets this criterion, we can proceed to the next step: finding the formula for the inverse function.
Finding the Inverse Function
Finding the inverse of a one-to-one function involves reversing the roles of the input and output. In essence, we want to find a function, denoted as f⁻¹(x), that "undoes" what the original function f(x) does. If f(a) = b, then f⁻¹(b) = a. This reversal is the core concept behind finding inverse functions.
The process of finding the inverse function typically involves the following steps. First, we replace f(x) with y to simplify the notation. Second, we swap x and y, reflecting the reversal of input and output. Third, we solve the resulting equation for y. This new expression for y will be the inverse function, f⁻¹(x). Finally, we replace y with f⁻¹(x) to denote the inverse function explicitly.
Let's apply these steps to our example function, f(x) = 7/(x+9).
- Replace f(x) with y:
y = 7/(x+9)
- Swap x and y:
x = 7/(y+9)
- Solve for y:
To solve for y, we first multiply both sides by (y+9) to eliminate the fraction:
x(y+9) = 7
Next, we distribute x on the left side:
xy + 9x = 7
Now, we isolate the term containing y by subtracting 9x from both sides:
xy = 7 - 9x
Finally, we divide both sides by x to solve for y:
y = (7 - 9x) / x
- Replace y with f⁻¹(x):
f⁻¹(x) = (7 - 9x) / x
Thus, the inverse function of f(x) = 7/(x+9) is f⁻¹(x) = (7 - 9x) / x. This formula allows us to reverse the mapping of the original function. For example, if we input a value into f(x) and then input the result into f⁻¹(x), we should obtain the original input value. This property serves as a check for the correctness of our inverse function.
Step-by-Step Calculation of the Inverse Function for f(x) = 7/(x+9)
To find the inverse function for f(x) = 7/(x+9), we follow a systematic approach that involves several key steps. This process ensures that we correctly reverse the roles of input and output, ultimately leading to the formula for the inverse function.
Step 1: Replace f(x) with y
The first step is to replace the function notation f(x) with the variable y. This simplifies the equation and makes it easier to manipulate. So, we rewrite the function as:
y = 7/(x+9)
Step 2: Swap x and y
The next crucial step is to swap the variables x and y. This reflects the fundamental idea of an inverse function, which is to reverse the roles of input and output. Swapping x and y gives us:
x = 7/(y+9)
Step 3: Solve for y
Now, our goal is to isolate y and express it in terms of x. This involves a series of algebraic manipulations. First, we multiply both sides of the equation by (y+9) to eliminate the fraction:
x(y+9) = 7
Next, we distribute x on the left side of the equation:
xy + 9x = 7
To isolate the term containing y, we subtract 9x from both sides:
xy = 7 - 9x
Finally, we divide both sides by x to solve for y:
y = (7 - 9x) / x
Step 4: Replace y with f⁻¹(x)
The last step is to replace y with the inverse function notation f⁻¹(x). This gives us the explicit formula for the inverse function:
f⁻¹(x) = (7 - 9x) / x
Therefore, the inverse function of f(x) = 7/(x+9) is f⁻¹(x) = (7 - 9x) / x. Each step in this process is essential to ensure the accurate derivation of the inverse function. This methodical approach allows us to reverse the mapping of the original function and express the inverse relationship explicitly.
The Inverse Function: f⁻¹(x) = (7 - 9x) / x
Having gone through the process of determining that f(x) = 7/(x+9) is a one-to-one function and subsequently calculating its inverse, we have arrived at the formula for the inverse function: f⁻¹(x) = (7 - 9x) / x. This function effectively reverses the operation of the original function, mapping outputs back to their corresponding inputs.
The inverse function, f⁻¹(x) = (7 - 9x) / x, provides a way to find the input value that produces a specific output in the original function. For instance, if we have an output value from f(x), we can input that value into f⁻¹(x) to obtain the original input. This property is a fundamental characteristic of inverse functions and can be used to verify the correctness of our calculation.
To further illustrate the relationship between the original function and its inverse, let's consider an example. Suppose we have an input value of x = 1 for the original function f(x) = 7/(x+9). Plugging this into f(x), we get:
f(1) = 7/(1+9) = 7/10
Now, if we input this result, 7/10, into the inverse function f⁻¹(x) = (7 - 9x) / x, we should obtain the original input value, 1. Let's check:
f⁻¹(7/10) = (7 - 9(7/10)) / (7/10)
To simplify this expression, we first find a common denominator for the terms in the numerator:
f⁻¹(7/10) = (70/10 - 63/10) / (7/10)
This simplifies to:
f⁻¹(7/10) = (7/10) / (7/10)
Dividing the fractions, we get:
f⁻¹(7/10) = 1
As expected, inputting the output of f(x) into f⁻¹(x) gives us the original input value, 1. This confirms that our calculated inverse function is indeed correct. The inverse function allows us to reverse the mapping of the original function, providing a tool for solving equations and understanding the relationships between inputs and outputs. The formula f⁻¹(x) = (7 - 9x) / x is a concise representation of this inverse relationship for the function f(x) = 7/(x+9).
Conclusion
In conclusion, we have successfully determined that the function f(x) = 7/(x+9) is one-to-one using the algebraic method and derived its inverse function, f⁻¹(x) = (7 - 9x) / x. The process involved verifying the one-to-one property by showing that f(x₁) = f(x₂) implies x₁ = x₂, and then systematically finding the inverse function by swapping x and y and solving for y. Understanding these steps is crucial for handling various functions and their inverses in mathematics. The inverse function provides a way to reverse the mapping of the original function, allowing us to find the input that corresponds to a given output. This concept is fundamental in many areas of mathematics and its applications.
