Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine algebraically whether the function is one-to-one. Verify your answer graphically. If the function is one-to-one, find its inverse. $$f(x)=\frac{1}{x^{2}}$$

Step-by-Step Solution

Answer

The function \(f(x) = \frac{1}{x^{2}}\) is not one-to-one and hence does not have an inverse.

Step 1: Algebraic Verification

To determine if the function is one-to-one algebraically, set \(f(x1) = f(x2)\) and solve for x.\n\(\frac{1}{x1^{2}} = \frac{1}{x2^{2}}\)\nCross multiply to obtain: \(x1^{2} = x2^{2}\)\nTaking root of both sides, we get:\(x1 = ±x2\). And since x1 = x2 and x1 = -x2, the function is not one-to-one.

Step 2: Graphical Verification

The graph of \(f(x) = \frac{1}{x^{2}}\) is an upward opening parabola. It is clear that for a given value of 'y', we get two values of 'x' (one positive and one negative). Thus, the function fails the horizontal line test, confirming it is not one-to-one.

Step 3: Finding the Inverse

Since the function is not one-to-one it does not have an inverse.

Key Concepts

Algebraic Verification

Understanding whether a function is one-to-one is crucial in algebra. This property ensures that each input of a function is paired with a unique output. Algebraic verification is a method to test this by setting two different inputs equal to each other and solving for the variables. For example, if we consider the function given in the exercise, we first equate the outputs for two different inputs, saying ewline \(f(x_1) = f(x_2)\). In the case of ewline \(f(x)=\frac{1}{x^2}\), this process involves setting ewline \(\frac{1}{x_1^2} = \frac{1}{x_2^2}\) and simplifying. When we solve this, we see that ewline \(x_1^2 = x_2^2\), which further leads to ewline \(x_1 = \pm x_2\). This implies that there are at least two different values (ewline \(x_1\) andewline \( -x_1\)) that can result in the same output. Therefore, the function fails the test for being one-to-one, as one output is not exclusively linked to a single input.

Graphical Verification

Graphical verification offers a visual way to understand the behavior of a function and is a great tool to determine if a function is one-to-one. For a function to be considered one-to-one, it must pass what is known as the horizontal line test—no horizontal line should intersect the graph of the function more than once. If it does, the function is not one-to-one.In the function ewline \(f(x) = \frac{1}{x^2}\), the graph plotted is an upward opening parabola that is symmetrical across the y-axis. This symmetry suggests that for any given value of ‘y’ above the vertex, there are two corresponding ‘x’ values (one positive and one negative). In layman's terms, this means for some outputs, you have more than one possible input. Through the graphical approach, it’s clear that the function fails to satisfy the horizontal line test, thus reaffirming that it is not one-to-one.

Function Inverse

The inverse of a function is essentially the function flipped or reversed, where the roles of inputs and outputs are switched. For a function to have an inverse, it needs to be one-to-one because this guarantees that each output comes from a unique input. Without this one-to-one property, you could end up with an inverse that doesn’t represent a function because a single input could provide two different outputs—which contradicts the definition of a function.When a function is not one-to-one, as we’ve shown algebraically and graphically for ewline \(f(x) = \frac{1}{x^2}\), finding an inverse is not possible. This is because the 'flipping' operation would result in a relation that doesn’t meet the criteria of a function. To reiterate, without the function being one-to-one, any potential inverse cannot assure a unique input for every output, rendering the concept of an inverse inapplicable in this case.