Problem 32
Question
Determine algebraically whether the function is one-to-one. If it is, find its inverse function. Verify your answer graphically. $$f(x)=\frac{x-4}{5}$$
Step-by-Step Solution
Verified Answer
The function \(f(x) = \frac{x-4}{5}\) is one-to-one, as shown algebraically. Its inverse function is \(f^{-1}(x) = 5x + 4\) which can be verified by graphical representation.
1Step 1: Determine if the function is one-to-one
Firstly, suppose that there are two inputs \(x1\) and \(x2\) such that \(f(x1) = f(x2)\). If the function is one-to-one, it should hold that \(x1 = x2\). So, suppose \(f(x1) = \frac{x1-4}{5} = f(x2) = \frac{x2-4}{5}\) Rearrange to get \(x1 - 4 = x2 - 4\) which simplifies to \(x1 = x2\). This proves algebraically that the given function is one-to-one.
2Step 2: Find the inverse function
To find the inverse function, use the original equation \(f(x) = \frac{x-4}{5}\) and solve for x. Start by replacing \(f(x)\) with \(y\). Next, swap the roles of x and y to get \(x = \frac{y-4}{5}\). Solving for y yields the inverse function \(f^{-1}(x) = 5x + 4\).
3Step 3: Graphical representation
On a graph, plot both \(f(x) = \frac{x-4}{5}\) and its inverse \(f^{-1}(x) = 5x + 4\). If the graph of the function and its inverse are reflections of each other in the line \(y = x\), this verifies the solution graphically.
Key Concepts
Determining one-to-one functionsFinding inverse functions algebraicallyGraphical verification of inverses
Determining one-to-one functions
A function is considered one-to-one (injective), if every element of the range is mapped to by exactly one element of the domain. Put simply, a one-to-one function never assigns the same value to two different domain elements.
To determine algebraically whether a function is one-to-one, we can employ a test. Let's take two hypothetical inputs, say, \(x_1\) and \(x_2\). Assign these inputs to the function and set them equal to each other, i.e., \(f(x_1) = f(x_2)\). If we can deduce from this equation that \(x_1 = x_2\), then the function passes the test for being one-to-one. In our exercise, the given function \(f(x) = \frac{x-4}{5}\) easily passes this test, as the algebraic manipulation shows that \(x_1 = x_2\), confirming its injective nature.
To determine algebraically whether a function is one-to-one, we can employ a test. Let's take two hypothetical inputs, say, \(x_1\) and \(x_2\). Assign these inputs to the function and set them equal to each other, i.e., \(f(x_1) = f(x_2)\). If we can deduce from this equation that \(x_1 = x_2\), then the function passes the test for being one-to-one. In our exercise, the given function \(f(x) = \frac{x-4}{5}\) easily passes this test, as the algebraic manipulation shows that \(x_1 = x_2\), confirming its injective nature.
Finding inverse functions algebraically
To find an inverse function algebraically, we need to perform a series of operations that essentially 'undo' the action of the original function. The first step is to write the function as an equation, with \(y\) representing the function's output, that is, \(y = f(x)\). Next, we switch \(x\) and \(y\) to reflect the inverse relationship. After swapping, our goal is to solve this new equation for \(y\).
This process may involve steps such as isolating the variable on one side of the equation and performing necessary arithmetic operations. In our case, after exchanging \(x\) and \(y\), we have \(x = \frac{y-4}{5}\). Multiplying both sides by 5 and then adding 4, we obtain the inverse function \(f^{-1}(x) = 5x + 4\). This new function will reverse the effect of our original function, essentially taking the output back to its initial input value.
This process may involve steps such as isolating the variable on one side of the equation and performing necessary arithmetic operations. In our case, after exchanging \(x\) and \(y\), we have \(x = \frac{y-4}{5}\). Multiplying both sides by 5 and then adding 4, we obtain the inverse function \(f^{-1}(x) = 5x + 4\). This new function will reverse the effect of our original function, essentially taking the output back to its initial input value.
Graphical verification of inverses
Graphically, a function and its inverse are reflections of each other over the line \(y = x\). This property serves as a visual verification of our work. To verify inverses graphically, we can plot both the original function and its proposed inverse on the same set of axes.
For the function \(f(x) = \frac{x-4}{5}\) and its inverse \(f^{-1}(x) = 5x + 4\), plotting both will allow us to observe if they are indeed reflections across the identity line. If they do mirror each other, this supports our algebraic conclusion that we have correctly determined the inverse. This visual confirmation is not only satisfying but also a helpful, immediate check to validate our inverse function.
For the function \(f(x) = \frac{x-4}{5}\) and its inverse \(f^{-1}(x) = 5x + 4\), plotting both will allow us to observe if they are indeed reflections across the identity line. If they do mirror each other, this supports our algebraic conclusion that we have correctly determined the inverse. This visual confirmation is not only satisfying but also a helpful, immediate check to validate our inverse function.
Other exercises in this chapter
Problem 31
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