Problem 31 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine algebraically whether the function is one-to-one. If it is, find its inverse function. Verify your answer graphically. $$f(x)=2 x+5$$

Step-by-Step Solution

Answer

The function \(f(x) = 2x + 5\) is indeed one-to-one. Its inverse function is \(f^{-1}(x) = (x-5)/2\). Graphical verification also confirms both these characteristics.

Step 1: Determining whether the function is one-to-one

A function is one-to-one if every y value corresponds to exactly one x value. For the function \(f(x) = 2x + 5\), we can see that for different x values, we will have different y values. Thus, the function is one-to-one.

Step 2: Finding the inverse of the function

To find the inverse of a function, we interchange the x and the y variables in the function and then solve for y. Thus, replacing \(f(x)\) by y in our original function gives us \( y=2x+5\). Interchanging x and y gives us \(x=2y+5\). Solving for y gives us \(y= (x-5)/2\). Thus, the inverse function \(f^{-1}(x) = (x-5)/2\).

Step 3: Verify the oneness and the inverse function graphically

Plot the function, its inverse, and the line \(y = x\). The function and its inverse should be reflections over the line \(y =x\). In other words, the graph of the function and its inverse should be symmetrical about the line \(y = x\).

Key Concepts

Finding Inverse Functions

Understanding the process of finding inverse functions is crucial for students as it not only solidifies their grasp of function operations but also aids in the exploration of symmetrical relationships in mathematics. Let's decode how to unravel an inverse function using a textbook example: if you're given the function f(x) = 2x + 5, the goal is to find an output's original input. To begin, replace f(x) with y, yielding y = 2x + 5. Afterward, swap the positions of x and y. Solving for y, you then isolate the variable to get y = (x - 5)/2, which gives you the inverse function f-1(x) = (x - 5)/2.

The trick lies in switching the input with the output, a key step signifying that the inverse function essentially undoes the original function's effect. However, not all functions have inverses, which is why it is imperative to first verify that a function is one-to-one before finding its inverse.

One-To-One Function Verification

One-to-one functions possess a distinctive characteristic where each element of the range is paired with exactly one element of the domain. In layman's terms, this means that no two different inputs will lead to the same output. To verify whether a given function is one-to-one, we employ various methods. For instance, the given function f(x) = 2x + 5 can be tested by recognizing that for any two distinct values of x, the resulting y values will also be distinct.

Horizontal Line Test

One illustrative method is the 'horizontal line test'. If any horizontal line intersects the graph of the function at no more than one point, the function is one-to-one. This graphical approach complements the algebraic understanding and ensures that students can verify a function's uniqueness both visually and through calculation.

Graphical Verification of Inverse Functions

Graphical verification serves as an excellent visual confirmation of one-to-one functions and their inverses. After determining that the function f(x) = 2x + 5 is indeed one-to-one and finding its inverse f-1(x) = (x - 5)/2, we plot both functions on the same set of axes. An essential reference is the line y = x, which acts as a mirror.

When plotted, the graph of the original function f(x) and the graph of the inverse f-1(x) should reflect across this line. This symmetrical relationship visually affirms the correctness of the inverse: if you 'fold' the graph along the line y = x, the function and its inverse would match up perfectly. Encouraging students to draw these graphs not only enhances their comprehension of the concept but also solidifies the relationship between a function and its inverse.