Problem 62 - Algebra and Trigonometry

Question

A one-to-one function is given. (a) Find the inverse of the function. (b) Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line \(y=x\). \(f(x)=2-\frac{1}{2} x\)

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = 4 - 2x\).

Step 1: Set the Function Equal to y

First, express the function in terms of \(y\). The original function is \(f(x) = 2 - \frac{1}{2}x\). Set this equal to \(y\): \[ y = 2 - \frac{1}{2}x \]

Step 2: Swap Variables

To find the inverse, swap the roles of \(x\) and \(y\). This means we replace \(y\) with \(x\) and \(x\) with \(y\):\[ x = 2 - \frac{1}{2}y \]

Step 3: Solve for y

Solve the equation from Step 2 for \(y\). First, isolate the term containing \(y\):\[ \frac{1}{2}y = 2 - x \]Multiply both sides by 2 to solve for \(y\):\[ y = 4 - 2x \]

Step 4: Write the Inverse Function

Write the inverse function with \(x\) as the independent variable, from the solved equation for \(y\):\[ f^{-1}(x) = 4 - 2x \]

Step 5: Verify the Graphical Reflection

To verify graphically, plot both \(f(x) = 2 - \frac{1}{2}x\) and its inverse \(f^{-1}(x) = 4 - 2x\) on the same screen. Also, plot the line \(y = x\). Check that the two functions are symmetric with respect to this line.

Key Concepts

One-to-One Functions

A one-to-one function is an essential concept in mathematics, particularly when discussing inverse functions. A function is considered one-to-one if each output value is paired with exactly one input value. This means that if the function maps two different inputs to the same output, it is not one-to-one. In simpler terms, a one-to-one function never "repeats" an output for a different input.
Why is this important? If a function is not one-to-one, then its inverse won't pass the Horizontal Line Test. The Horizontal Line Test confirms that no horizontal line drawn through the graph touches more than one point. If it does, the function is not one-to-one, and thus, it cannot have an inverse function that is also a function.
To check for one-to-one functionality in practice:

• Consider if each output is paired with a unique input.
• Use the Horizontal Line Test on the graph.

For example, the function given in our problem, \( f(x) = 2 - \frac{1}{2}x \), is one-to-one. The inverse function, \( f^{-1}(x) = 4 - 2x \), also works perfectly within this property.

Graphical Reflection

Graphical reflection is a visual method of understanding inverse functions. When you graph a function and its inverse, you will notice that they appear to "mirror" each other across the line \( y = x \). This line acts as a reflection line, and seeing both curves acting as mirrors verifies their inverse relationship visually.
In the example given:

• The function \( f(x) = 2 - \frac{1}{2}x \)
• Its inverse \( f^{-1}(x) = 4 - 2x \)

When graphed, these functions should show symmetry about the line \( y = x \). If any point \( (a, b) \) lies on the graph of the original function, then the point \( (b, a) \) should lie on the graph of its inverse.
The graphical reflection thus serves as both a verification method for correctly identifying inverse functions and an excellent illustration of the concept of inverse functions in mathematics.

Line Symmetry

Line symmetry in the context of functions refers to the idea that parts of a function graph can be flipped over a given line and still appear identical. This is especially relevant when discussing inverse functions, where the line of symmetry is typically the line \( y = x \).
By using line symmetry, one can confirm that a function and its inverse are correctly derived. When a function \( f(x) \) reflects over the line \( y = x \), its symmetrical properties allow any point \( (x, y) \) on the curve of \( f(x) \) to have a corresponding point \( (y, x) \) on \( f^{-1}(x) \).
Practically, this is how you can verify that the inverse has been calculated right:

• Draw the line of symmetry, \( y = x \).
• Plot both the original function and the inverse on the same axes.
• Ensure each point \( (x, y) \) on \( f(x) \) has a counterpart \( (y, x) \) on \( f^{-1}(x) \).

Only functions that exhibit line symmetry in this way about \( y = x \) can be considered accurately paired as inverse functions.