Problem 42 - Contemporary Precalculus

Question

None of the functions has an inverse. State at least one way of restricting the domain of the function (that is, find a function with the same rule and a smaller domain so that the restricted function has an inverse. Then find the rule of the inverse function. Example: \(f(x)=x^{2}\) has no inverse. But the function \(h\) with domain all \(x \geq 0\) and rule \(h(x)=x^{2}\) is increasing (its graph is the right half of the graph of \(f-\) see Figure \(2-2\) on page 78 )and therefore has an inverse. $$f(x)=\sqrt{4-x^{2}}$$

Step-by-Step Solution

Answer

Answer: The inverse function of the restricted \(f(x)\) is \(f^{-1}(x) = \sqrt{4-x^2}\), with the domain \(0\leq x\leq 2\) and range \(0\leq y\leq 2\).

Step 1: Determine the domain of the function

The given function is \(f(x) = \sqrt{4-x^2}\). To find the domain, we need to make sure the value inside the square root is greater than or equal to 0: \(4-x^2\geq 0\). We can factor this inequality as follows: \((2-x)(2+x)\geq 0\). This gives us the domain of \(f(x): -2\leq x\leq 2\).

Step 2: Restrict the domain to make the function increasing or decreasing

We want to choose a smaller domain to ensure that the function is either increasing or decreasing. Since the function is symmetrical along the y-axis (even function), we can choose either the positive or negative side of the domain for our restriction. To keep things simple, let's restrict the function to the right half of the graph, where \(0\leq x\leq 2\). This will make our function increasing in the restricted domain.

Step 3: Invert the restricted function

Now let's find the inverse function of \(f(x)\) in the restricted domain, \(0\leq x\leq 2\). To do this, follow these steps: 1. Replace \(f(x)\) with \(y\): \(y = \sqrt{4-x^2}\). 2. Swap \(x\) and \(y\): \(x = \sqrt{4-y^2}\). 3. Solve for \(y\). We can first square both sides to get rid of the square root: \(x^2 = 4-y^2\). 4. Add \(y^2\) and subtract \(x^2\) to both sides: \(y^2 = 4-x^2\). 5. Finally, express \(y\) in terms of \(x\): \(y = \sqrt{4-x^2}\). The inverse function of the restricted \(f(x)\) is \(f^{-1}(x) = \sqrt{4-x^2}\), with the domain \(0\leq x\leq 2\) and range \(0\leq y\leq 2\).

Key Concepts

Domain Restriction

Understanding Domain Restriction is vital when working with functions that don't naturally have inverses. In precalculus, when we encounter a function like f(x) = sqrt(4-x^2), we observe how it behaves within its original domain. If the function's graph isn't one-to-one, as in this case where it's a semi-circle, it won't have an inverse across the entire domain because it doesn't pass the horizontal line test.

To obtain an invertible function, we limit the domain, so the function is either strictly increasing or decreasing. In our exercise, restricting the domain of f(x) to [0, 2] ensures it becomes an increasing function. By constraining the values of x to non-negative numbers, the inverse can be uniquely determined for each input within this new domain.

Increasing Functions

An Increasing Function is one where the output rises as the input increases. In graphical terms, if any vertical line intersects the curve at most once, it is strictly increasing. In the context of our exercise, we aim to find a domain in which f(x) = sqrt(4-x^2) is increasing. By limiting x between 0 and 2, we ensure that no horizontal line intersects the graph more than once within this interval. This guarantees that each x has a unique y value, meeting the criteria for a function to have an inverse. This characteristic not only simplifies finding the inverse but also clarifies the concept for students—each input has a single, specific output within the restricted domain.

Finding Inverse Functions

The key steps in Finding Inverse Functions involve swapping the x and y variables and solving for the new y. This process turns the outputs of the original function into the inputs of the inverse function, and vice versa. From our exercise, after applying the domain restriction, we start by replacing f(x) with y and then switch x and y to find the inverse. The final expression, f^{-1}(x) = sqrt(4-x^2), defines the inverse function within the restricted domain. The inverse function reflects across the line y=x, and this symmetry is crucial for understanding the relationship between a function and its inverse.

Square Root Functions

A Square Root Function typically looks like f(x) = sqrt(x) and involves taking the square root of the input x. Regarding our problem, f(x) = sqrt(4-x^2) is a modified square root function. It's paramount to remember that square root functions are not one-to-one on their entire domain since they produce both positive and negative results for every positive input.

Therefore, we must carefully consider the domain to define a function that is either entirely non-negative or non-positive, to make it suitable for finding an inverse. In our specific case, by ensuring the domain is restricted to [0, 2], we are considering only the non-negative values of x, which is crucial for establishing a function that produces unique outputs and thus has an inverse.