Question

Find the inverse of \(f .\) Then sketch the graphs of \(f\) and \(f^{-1}\) on the same set of axes. $$ f(x)=\sin (2 x-1), \quad \frac{1}{2}\left(1-\frac{\pi}{2}\right) \leq x \leq \frac{1}{2}\left(1+\frac{\pi}{2}\right) $$

Short answer

The inverse of the given function is \(f^{-1}(x) = \frac{1}{2}(\arcsin(x) + 1)\), with a domain of \(-1 \leq x \leq 1\). To sketch the graphs of \(f(x) = \sin(2x - 1)\) and \(f^{-1}(x) = \frac{1}{2}(\arcsin(x) + 1)\) on the same set of axes, remember that: 1. The original function, \(f(x)\), is a horizontally compressed and shifted sine function. 2. The inverse function, \(f^{-1}(x)\), is a vertically compressed and shifted arcsine function. Make sure to label the functions and their domains and ranges. The inverse function's graph should appear as a reflection of the original function's graph over the line \(y = x\).

Step-by-step solution

Determine the Range

To find the inverse of the function, we need to first find the range of the function by looking at the function within the given domain. \(\sin(x)\) has a range of \([-1, 1]\) for any interval of length greater than or equal to \(2\pi\), and since our interval spans \(\pi\), it will cover the entire range. So, \(f(x)\) will have a range of: \( -1 \leq \sin(2x-1) \leq 1 \)

Find the inverse function \(f^{-1}(x)\)

Next, we find the inverse function, \(f^{-1}(x)\). We start by replacing \(f(x)\) with \(y\): \(y =\sin(2x-1)\) Now we swap all instances of \(x\) and \(y\): \(x =\sin(2y-1)\) Finally, we solve for \(y\) to give us the inverse function, \( y = \frac{1}{2}(\arcsin(x) + 1) \) Thus, the inverse function is: \( f^{-1}(x) = \frac{1}{2}(\arcsin(x)+1) \)

Determine the restricted domain of the inverse function

To sketch the graph of the inverse function, we also need its domain. Since the range of \(f(x)\) is \([-1, 1]\), the domain of the inverse function will be: \( -1 \leq x \leq 1 \)

Sketch the graphs of both the original and the inverse function

Now, it's time to sketch the graphs of both the original and the inverse function on the same set of axes. 1. The original function, \(f(x) = \sin(2x-1)\), is a sine function with a horizontal shift of \(\frac{1}{2}\) to the right and a horizontal compression by a factor of \(2\). This means the graph will oscillate twice as fast as the regular sine function. 2. The inverse function, \(f^{-1}(x) = \frac{1}{2}(\arcsin(x)+1)\), is an arcsine function with a vertical shift of \(\frac{1}{2}\) upwards and a vertical compression by a factor of \(2\). This means the graph will look like a usual arcsine function, but half as tall and shifted upwards. We can sketch these functions on the same set of axes, considering the given domain and range. Make sure to label the functions and their domains and ranges. The inverse function should appear to be a reflection of the original function over the line \(y = x\).

Key concepts

Finding Inverse Functions

Inverse functions can be thought of as a 'mirror image' in mathematics. When we find the inverse of a function, we're essentially figuring out the original input given an output. It's like retracing your steps in a maze to find your way back to the start.

The process involves a few steps. Switching the roles of the input and output variables is the initial move. In more formal terms, replace the function's output variable with 'y' and then interchange 'x' and 'y'. This swapping hints at what the inverse will do - it'll take the outputs of the original function and retrieve the corresponding inputs.

Next, solve the resulting equation for the newly placed 'y' to express the inverse relationship. The calculation might require algebraic maneuvers, sometimes involving solving for a variable within a trigonometric function. Once we have 'y' isolated, we get the inverse function, usually denoted as \(f^{-1}(x)\).

However, not all functions have an inverse. To have an inverse, a function must be bijective - meaning it's both one-to-one (each input has a unique output) and onto (covers the entire possible range of outputs). Graphically, if a function passes the Horizontal Line Test (no horizontal line intersects the graph more than once), it has an inverse.

Graphing Sine Functions

Sine functions are like the heartbeat of mathematics, with a consistent ebb and flow that repeats over intervals. Graphing a sine function involves understanding its amplitude (height), period (width of a single cycle), phase shift (horizontal movement), and vertical shift.

Standard sine functions take the form \(y = \sin(x)\), oscillating between 1 and -1, with a period of \(2\pi\). However, they can be manipulated. For instance, \(f(x) = \sin(2x-1)\) has a period halved to \(\pi\) because of the coefficient 2, which makes the function oscillate faster. The subtraction of 1 shifts the graph to the right, as it indicates that the usual starting point of 0 (where the sine wave crosses the x-axis) is now moved.

Drawing the Graph

Begin by marking the maximum and minimum values, the x-axis crossings, and noting how many cycles will fit within the given domain. Use the transformed sine function to pinpoint these key moments accurately. The resulting graph showcases the beautifully rhythmic nature of trigonometric functions, and understanding these transformations allows us to graph them effectively on a set of axes.

Arcsin(x) Properties

Speaking of inverse trigonometric functions, arcsine – often denoted as \(\arcsin(x)\) or \(\sin^{-1}(x)\) – is the reverse of the sine function. It takes an output value and tells us the original angle (or input) for which the sine gave that value. But here's the catch: because the sine function is not one-to-one (it doesn't pass the Horizontal Line Test), we must restrict its domain to make it invertible.

The domain of \(\arcsin(x)\) is limited to \([-1, 1]\), and its range to \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) to ensure it's a function (each x-value gives us just one y-value). In practice, this means \(\arcsin(x)\) will only provide an angle between -90° and 90°, inclusive. This restricted view allows us to have a clear, unambiguous inverse function for sine. Also, \(\arcsin(x)\) bears the hallmarks of any good inverse function: if you were to graph both the sine function (within its restricted domain) and \(\arcsin(x)\) on the same axes, they would reflect across the line \(y = x\).

Domain and Range of Functions

Every function attends a fancy party where the guests are inputs and ouputs. The 'domain' is the guest list of possible inputs, and the 'range' is that of the possible outputs. For the party hosted by our trigonometric functions, the domain is the set of permissible angles, and the range is the set of sine values those angles can produce.

In our function \(f(x) = \sin(2x-1)\), the domain is confined within \(\frac{1}{2}(1-\frac{\pi}{2}) \leq x \leq \frac{1}{2}(1+\frac{\pi}{2})\), which means we only care about the sine values where 'x' is within that interval. For the inverse \(f^{-1}(x)\), the domain is now the range of the original function, which is \( [-1, 1] \), since that's where the sine function's outputs lie.

Understanding domain and range helps us form a complete picture of a function - we know where it starts, where it can go, and importantly, where it stops. These concepts are vital for graphing, as they guide us in setting up the axes' limits and knowing which parts of the function to draw.