Problem 51
Question
Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=\frac{1}{x}$$
Step-by-Step Solution
Verified Answer
The inverse of the function \(f(x) = \frac{1}{x}\) is \(f^{-1}(x) = \frac{1}{x}\). The graph of the function and its inverse coincide as the function is self-inverse.
1Step 1: Identify the Function
The original function given is \(f(x)=\frac{1}{x}\). This function is known as a hyperbolic function.
2Step 2: Find the Inverse of the function
To find the inverse of a function, swap the y and x. So if \(y = \frac{1}{x}\) then the inverse is \(x = \frac{1}{y}\). Solve this to find the inverse function, which is \(f^{-1}(x) = \frac{1}{x}\). So, the function is self-inverse.
3Step 3: Plot the original function and its inverse
Plot the function \(f(x) = \frac{1}{x}\) and its inverse \(f^{-1}(x) = \frac{1}{x}\) on the same set of axes. Since the function is self-inverse, the two plots will coincide.
Key Concepts
Inverse FunctionsHyperbolic FunctionsGraphing Functions
Inverse Functions
Understanding inverse functions is crucial in algebra and calculus. In essence, finding an inverse involves swapping the input (usually x) with the output (usually y), and solving for the new output in terms of the new input. Formally, for a function \(f(x)\), the inverse \(f^{-1}(x)\) means that if \(f(a) = b\), then \(f^{-1}(b) = a\).
In the exercise, the given function is a simple rational function, \(f(x) = \frac{1}{x}\), which interestingly is its own inverse. Let's dive a bit deeper into this self-inverse characteristic.
When a function is its own inverse, applying it twice will yield the original value you started with. Mathematically, that means \(f(f(x))=x\) for all x in the function's domain. For our function, if you substitute \(\frac{1}{x}\) into itself, you indeed get back to x, reinforcing the concept of a function being self-inverse.
In the exercise, the given function is a simple rational function, \(f(x) = \frac{1}{x}\), which interestingly is its own inverse. Let's dive a bit deeper into this self-inverse characteristic.
When a function is its own inverse, applying it twice will yield the original value you started with. Mathematically, that means \(f(f(x))=x\) for all x in the function's domain. For our function, if you substitute \(\frac{1}{x}\) into itself, you indeed get back to x, reinforcing the concept of a function being self-inverse.
Hyperbolic Functions
Hyperbolic functions, which include \(sinh\), \(cosh\), and \(tanh\), are analogs of the trigonometric functions but for a hyperbola rather than a circle. However, the function from the exercise, \(f(x) = \frac{1}{x}\), is a hyperbolic function in a more general sense. The term 'hyperbolic' implies that the graph of the function will produce a hyperbola, a curve with two branches.
The graph of \(f(x) = \frac{1}{x}\) indeed forms a hyperbola that approaches both the x-axis and y-axis but never touches them - these axes are termed asymptotes. Hyperbolic functions like these are important in many areas of science and engineering, often appearing in situations where there is rapid growth or decay, such as in certain models of population growth or radioactive decay.
The graph of \(f(x) = \frac{1}{x}\) indeed forms a hyperbola that approaches both the x-axis and y-axis but never touches them - these axes are termed asymptotes. Hyperbolic functions like these are important in many areas of science and engineering, often appearing in situations where there is rapid growth or decay, such as in certain models of population growth or radioactive decay.
Graphing Functions
Graphing is a visual way of representing a function's behavior on a set of axes. For the function \(f(x) = \frac{1}{x}\), plotting points can help visualize the resulting hyperbola. Important things to consider when graphing include:
Because \(f(x)\) and \(f^{-1}(x)\) are the same in this scenario, the graph will show a symmetric pattern about the line \(y = x\). Each point on the graph of \(f(x)\) has a corresponding point on \(f^{-1}(x)\) that is its mirror image across the line \(y = x\). In other exercises with different functions, plotting both on the same axes can beautifully illustrate the 'undoing' nature of inverse functions through their reflected positions across this line.
- The shape of the graph (e.g., line, curve, hyperbola).
- Key points, such as intercepts with axes and turning points.
- Behavior as x approaches certain values, like asymptotes.
Because \(f(x)\) and \(f^{-1}(x)\) are the same in this scenario, the graph will show a symmetric pattern about the line \(y = x\). Each point on the graph of \(f(x)\) has a corresponding point on \(f^{-1}(x)\) that is its mirror image across the line \(y = x\). In other exercises with different functions, plotting both on the same axes can beautifully illustrate the 'undoing' nature of inverse functions through their reflected positions across this line.
Other exercises in this chapter
Problem 50
In Exercises \(47-52,\) let \(b=\log\) k. Write each expression in terms of b. Assume \(k>0\). $$\log k^{4}$$
View solution Problem 50
Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{7} 230$$
View solution Problem 51
Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log (3 x+1)-\log \left(x^{2}+1\right)=0$$
View solution Problem 51
In Exercises \(47-52,\) let \(b=\log\) k. Write each expression in terms of b. Assume \(k>0\). $$\log \frac{1}{k}$$
View solution