Question

Sketch a function that is one-to-one and positive for \(x \geq 0 .\) Make a rough sketch of its inverse.

Short answer

Short Answer: An appropriate function to consider is the exponential function \(f(x) = e^x\), which is one-to-one and always positive for \(x \geq 0\). To sketch the inverse function, we swap the x and y coordinates of the points in the original function, resulting in points such as (1, 0), (\(e\), 1), and (\(1/e\), -1). By connecting these points with a smooth curve, we can sketch the inverse function, which is the natural logarithm function \(f^{-1}(x) = \ln{x}\). Both the original function and its inverse are positive and one-to-one for all \(x \geq 0\).

Step-by-step solution

Determine an appropriate function

Since we need a one-to-one and positive function for \(x \geq 0\), a simple choice to consider is the exponential function \(f(x) = e^x\). This function is one-to-one and always positive for all real numbers \(x \geq 0\).

Sketch the function

To sketch the graph of the function \(f(x) = e^x\), plot a few points to establish the general shape of the graph. Key points to consider are (0, 1), (1, \(e\)), and (-1, \(1/e\)). Connect these points with a smooth curve that continually increases as x-value progresses. The curve will approach but never touch the x-axis.

Sketching the inverse function

To find the inverse of the function, swap the x and y coordinates for each point on the original function. These new points represent the graph of the inverse function. For our points (0, 1), (1, \(e\)), and (-1, \(1/e\)), the swaps result in the following coordinates: 1. (1, 0) 2. (\(e\), 1) 3. (\(1/e\), -1) Now, plot these points on the same graph and connect them with a smooth curve. Like the original function, the inverse function will also continually increase as the x-value progresses. The curve will touch the x-axis at x = 1 but will not intersect the y-axis.

Reflect the inverse function across the line y=x

Reflecting the graph of the original function across the diagonal line y=x will create a rough sketch of the inverse function. The line y=x serves as a mirror to help visualize the reflection. Your final graph should show the original function \(f(x) = e^x\) and its inverse function, \(f^{-1}(x) = \ln{x}\). Both functions will be positive and one-to-one for all \(x \geq 0\).

Key concepts

Inverse Functions

Inverse functions are like mathematical mirrors. They "undo" what the original function does. To find the inverse of a function, you simply switch the roles of the input and output. In practical terms, you swap the \(x\) and \(y\) values in the function's equation.
For example, if you have a function \(f(x) = e^x\), its inverse, \(f^{-1}(x)\), will satisfy \(f(f^{-1}(x)) = x\) and vice versa.
For exponential functions like \(e^x\), the inverse is the natural logarithm function \(\ln{x}\). This means that while \(e^x\) exponentially increases as \(x\) increases, \(\ln{x}\) will logarithmically increase as \(x\) increases.

• To check if two functions are inverses, apply one function to the output of the other, you should end up back with the original \(x\).
• Graphically, the inverse function is a reflection across the line \(y=x\).

One-to-One Functions

Understanding one-to-one functions is crucial for discussing inverses. A function is called "one-to-one" if every output is the result of one and only one input.
Simply put, no two different inputs \(x_1\) and \(x_2\) will have the same output, meaning if \(f(x_1) = f(x_2)\), then \(x_1 = x_2\). This property ensures that inverse functions can exist because each output maps back to a single input.

• If a function is one-to-one, you can easily find its inverse.
• Commonly used tests are the horizontal line test on a graph, where no horizontal line should intersect the graph of the function more than once.
• Examples of one-to-one functions include linear functions \(f(x) = mx+b\) where \(meq 0\) and the exponential function \(f(x) = e^x\).

Graph Sketching

Sketching the graph of a function and its inverse helps to visualize the relationship between the two. When sketching \(f(x) = e^x\), identify key points such as (0, 1), (1, \(e\)), and (-1, \(1/e\)). These points help form the shape of the graph, which increases continuously as \(x\) increases.
For the inverse function \(f^{-1}(x) = \ln{x}\), swap the \(x\) and \(y\) coordinates of the exponential points. This results in points like (1, 0), (\(e\), 1), and (\(1/e\), -1).
To sketch, draw both functions on the same graph, reflecting the inverse function over the line \(y = x\). This line acts as a perfect mirror, ensuring symmetry between a function and its inverse.

• Both functions will never intersect the \(y\)-axis in the given domain.
• The inverse function approach becomes clear as the curve aligns its growth mirrored from its original function.