Problem 69 - College Algebra with Modeling and Visualization

Question

Restrict the domain of \(f(x)\) so that \(f\) is one to-one. Then find \(f^{-1}(x)\). Answers may vary. $$ f(x)=\sqrt{9-2 x^{2}} $$

Step-by-Step Solution

Answer

Restrict the domain to [0, \(\sqrt{\frac{9}{2}}\)] to make \(f(x)\) one-to-one; then, \(f^{-1}(x) = \sqrt{\frac{9 - x^2}{2}}\).

Step 1: Identify the Domain of the Original Function

The function given is \(f(x) = \sqrt{9 - 2x^2}\). Since we have a square root, the expression under the square root \(9 - 2x^2\) must be greater than or equal to zero. Set \(9 - 2x^2 \geq 0\) to find the domain in which the function is defined.

Step 2: Solve the Domain Inequality

Solve the inequality \(9 - 2x^2 \geq 0\) by first rewriting it as \(2x^2 \leq 9\). Divide through by 2 to obtain \(x^2 \leq \frac{9}{2}\). Take the square root of both sides to find \(-\sqrt{\frac{9}{2}} \leq x \leq \sqrt{\frac{9}{2}}\). Thus, the domain of the function is \([-\sqrt{\frac{9}{2}}, \sqrt{\frac{9}{2}}]\).

Step 3: Restrict the Domain to Make f One-to-One

To make \(f(x)\) one-to-one, we restrict the domain further. A typical restriction for square root functions is to either take the positive or negative half of the root. Choose the domain \([0, \sqrt{\frac{9}{2}}]\). Now \(f(x)\) is one-to-one because it is strictly decreasing over this domain.

Step 4: Find the Inverse Function

Begin by setting \( y = f(x) = \sqrt{9 - 2x^2}\) and solve for \(x\) in terms of \(y\). Square both sides to get \(y^2 = 9 - 2x^2\). Rearrange to find \(2x^2 = 9 - y^2\). Solve for \(x^2\) giving \(x^2 = \frac{9 - y^2}{2}\), and finally take the square root to find \(x\). Since we chose the domain \([0, \sqrt{\frac{9}{2}}]\), take the positive root: \(x = \sqrt{\frac{9 - y^2}{2}}\). Thus, \(f^{-1}(x) = \sqrt{\frac{9 - x^2}{2}}\) for the restricted domain.

Key Concepts

One-to-One Function

A one-to-one function, or injective function, is a type of function where each input corresponds to a unique output. This means that if you have two different inputs, they will produce two different outputs. This property is crucial when finding an inverse because it ensures that each value in the domain maps to one and only one value in the range, and vice versa.

• Understanding this means a function does not "repeat" any output for different inputs.
• For example, a line with a positive or negative slope is one-to-one, whereas a parabola opening upwards is not.

To determine if a function is one-to-one, you can apply the horizontal line test, which states that if any horizontal line crosses the graph more than once, the function is not one-to-one. In the context of inverse functions, only one-to-one functions have inverses that are also functions. Restricting a function's domain can often fix this if the initial function fails the test.

Domain Restriction

In mathematics, domain restriction refers to narrowing the set of input values that a function can accept. Often, this is necessary to transform a non-one-to-one function into a one-to-one.

• This is done by restricting the domain so that the function is either always increasing or always decreasing over this range.
• In our example, the function is originally defined over the domain \(-\sqrt{\frac{9}{2}}, \sqrt{\frac{9}{2}}\) but has been restricted to \[0, \sqrt{\frac{9}{2}}\] to make it one-to-one.
• The domain restriction ensures that each output is uniquely paired with exactly one input, which is vital for finding the inverse.

Domain restriction might feel like limiting a function, but it often simplifies it to make it more useful in different applications, such as solving equations or modeling certain behaviors.

Square Root Functions

Square root functions are a specific type of function defined by an expression under a square root, usually taking the form \(f(x) = \sqrt{x}\). These functions are only defined where the expression under the square root is greater than or equal to zero because square roots of negative numbers are not real in standard arithmetic.

• This constraint naturally limits the domain of square root functions since the values of \(x\) must ensure the expression is non-negative.
• Like other functions, square root functions can also be transformed and translated, but they generally have a starting point, or vertex, at the origin \((0, 0)\), extending horizontally in the positive direction.

Square root functions particularly highlight the need to consider domain and frequent restriction to accommodate one-to-one properties. In this case, transforming \( f(x) = \sqrt{9 - 2x^2} \), the domain restriction helps achieve the necessary conditions to derive its inverse.