Problem 72 - 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

Inverse function is \( f^{-1}(x) = 4 - 2x \). Graphs should reflect across \( y=x \).

Step 1: Express the function as an equation

Start with the given function and express it as an equation: \( y = 2 - \frac{1}{2}x \).

Step 2: Solve for x in terms of y

To find the inverse of the function, interchange the roles of \( x \) and \( y \) to solve for \( x: x = 2 - \frac{1}{2}y \).

Step 3: Isolate y

Rearrange the equation to solve for \( y \): \( \frac{1}{2}y = 2 - x \).

Step 4: Solve for y

Multiply both sides by 2 to make \( y \) the subject of the equation: \( y = 4 - 2x \). This is the inverse function \( f^{-1}(x) = 4 - 2x \).

Step 5: Graph the original and inverse functions

Graph the original function \( f(x) = 2 - \frac{1}{2}x \) and the inverse function \( f^{-1}(x) = 4 - 2x \) on the same set of axes. Also, draw the line \( y = x \) for reference.

Step 6: Verify reflection across y=x

Both the function and its inverse should be symmetric with respect to the line \( y = x \). This confirms that the functions are indeed inverses of each other.

Key Concepts

One-to-One Functions

A key characteristic of one-to-one functions is that each value of the domain (input) maps to a unique value in the range (output). In simpler terms, not only does each output correspond to exactly one input, but each input also corresponds to exactly one output. This ensures no duplication, which makes it possible to have an inverse function.

To determine if a function is one-to-one, one can use the horizontal line test. If any horizontal line intersects the function’s graph more than once, the function is not one-to-one. By passing this test, you confirm that the function has an inverse.

Understanding one-to-one functions is fundamental because only these types of functions can have inverses. When you know a function is one-to-one, finding its inverse involves switching the roles of the dependent and independent variables and solving for the new dependent variable.

Reflection Across y=x

When graphing both a function and its inverse, they will appear as mirror images across the line represented by the equation \(y = x\). This line is a diagonal line through the origin, where every point on the line has coordinates that are equal, like (1,1), (2,2), and so on.

Graphically, reflecting a point across this line swaps the x and y coordinates of the point. So, if a point on the original function is (a, b), its mirror image on the inverse will be (b, a).

This reflection property serves as a visual confirmation that two functions are truly inverses. If the graphs of the functions exhibit this symmetry about the \(y = x\) line, it confirms their inverses relationship. Thus, by graphing, we can easily verify the correctness of our work finding an inverse.

Graphing Functions

Graphing functions is a powerful tool for visualizing the relationship between variables. It allows you to see how the output changes with different inputs and is a vital skill in mathematics.

For the function \(f(x) = 2 - \frac{1}{2}x\), its graph is a straight line with a negative slope. This tells us the function decreases at a constant rate as \(x\) increases. The inverse function, \(f^{-1}(x) = 4 - 2x\), also graphs as a line but with a steeper negative slope, indicating it decreases more rapidly.

When graphing these functions, it is essential to also graph the line \(y = x\) as a reference. This helps in checking the symmetry of the graphs, confirming the inverse relationship visually through reflection. Graphing makes it easier to interpret the behavior and properties of functions and their inverses quickly.