Problem 110 - A Graphical Approach to College Algebra

Question

Use a graph with the given viewing window to decide which functions are one- to-one. If a function is one-to-one, give the equation of its inverse function. Check your work by graphing the inverse function on the same coordinate axes. $$f(x)=x^{4}-5 x^{2}+6 ;[-3,3] \text { by }[-1,8]$$

Step-by-Step Solution

Answer

The function is not one-to-one; it has no inverse over \([-3, 3]\).

Step 1: Determine One-to-One Using Horizontal Line Test

A function is one-to-one if any horizontal line drawn through its graph intersects it at most once. We will graph the function \( f(x) = x^4 - 5x^2 + 6 \) over the given interval \([-3, 3]\) and visually inspect it to see if any horizontal line cuts the graph in more than one place. If yes, then it's not one-to-one.

Step 2: Graph the Function

Plot the function \( f(x) = x^4 - 5x^2 + 6 \) using a graphing tool within the window \([-3, 3]\) for the x-axis and \([-1, 8]\) for the y-axis. Observing this graph will help confirm the number of intersection points for horizontal lines.

Step 3: Analyze the Graph

Observe the graph of \( f(x) = x^4 - 5x^2 + 6 \). Notice that it is symmetric and creates a 'W' shape, indicating there are multiple points at which horizontal lines can intersect more than once within the interval \([-3, 3]\).

Step 4: Conclusion on One-to-One Status

Since there are horizontal lines that intersect the graph at more than one point, the function \( f(x) = x^4 - 5x^2 + 6 \) is not one-to-one over the interval \([-3, 3]\). Thus, it does not have an inverse function on this interval.

Step 5: Verify Using an Attempt to Find the Inverse

An inverse function can only be derived algebraically if the original function is one-to-one, and since it's not, any attempt to do so will not result in a valid function across the interval. Therefore, attempting to verify by algebraic means confirms the lack of an inverse.

Key Concepts

Horizontal Line Test

The horizontal line test is a simple, yet effective method to determine if a function is one-to-one. Here's the big idea:- A function is deemed one-to-one if every horizontal line you could possibly draw intersects the graph at most once.- If you find even a single horizontal line that cuts the graph at two or more points, the function is not one-to-one.

How to Implement the Horizontal Line Test:- First, graph the function you’re interested in.- Then, imagine or draw horizontal lines and see how many times they touch the curve.For example, consider the function \( f(x) = x^4 - 5x^2 + 6 \) within the interval \([-3, 3]\). When graphed, this function forms a 'W' shape.- In this shape, numerous horizontal lines would intersect the graph twice.- This means the function is not one-to-one over this interval, indicating no unique inverse function exists.

Inverse Function

An inverse function essentially reverses the original function's operation. It "undoes" what the original function does:- If \( f(x) \) takes a number \( a \) to a number \( b \), then its inverse \( f^{-1}(x) \) takes \( b \) back to \( a \).- A one-to-one function always has an inverse.How to Determine if a Function Has an Inverse:- First, ensure the original function is one-to-one using the horizontal line test.- If it passes the test, you can proceed to find its inverse by swapping the roles of \( x \) and \( y \) and solving for \( x \).However, for the function \( f(x) = x^4 - 5x^2 + 6 \), since it fails the horizontal line test over the interval \([-3, 3]\), it does not have an inverse on this interval.- This means that you can't have a unique function which reversibly maps outputs back to inputs through this range.

Graphing Functions

Graphing is a powerful visual tool for understanding and analyzing functions. When dealing with complex expressions like \( f(x) = x^4 - 5x^2 + 6 \), you gain vital insights by plotting the function.Steps to Graph a Function:1. Choose a suitable graphing tool or software.2. Input the function expression: in our example, \( f(x) = x^4 - 5x^2 + 6 \).3. Select the correct viewing window. For this specific function, use \([-3, 3]\) for the x-range, and \([-1, 8]\) for the y-range.Once you have plotted the graph:- Observe the behavior and shape of the curve.- Look for any symmetry or repeating patterns which might contribute to analysis through tests like the horizontal line test.Graphing not only helps in determining one-to-one status but is also crucial for discovering if the inverse function plotting could be done if the original function were one-to-one.