Problem 75 - Algebra and Trigonometry

Question

The given function is not one-to-one. Restrict its domain so that the resulting function \(i s\) one-to-one. Find the inverse of the function with the restricted domain. (There is more than one correct answer.) $$ f(x)=4-x^{2} $$

Step-by-Step Solution

Answer

Restrict the domain to \(x \geq 0\). The inverse is \(f^{-1}(x) = \sqrt{4 - x}\), for \(x \leq 4\).

Step 1: Identify Function Features

The function given is a quadratic function, which is a parabola opening downwards. This can be seen from the negative sign in front of the \(x^2\) term.

Step 2: Find the Vertex

To find where the function is symmetric around, find the vertex by setting its derivative to zero: \(f'(x) = -2x = 0\). So, \(x = 0\). The vertex is at the point \((0, 4)\).

Step 3: Determine Suitable Domain

To make the function one-to-one, we can restrict the domain to either the left or the right of the vertex. Choose the domain where \(x \geq 0\) or \(x \leq 0\).

Step 4: Define Restricted Function

Let's restrict it to \(x \geq 0\). The function becomes \(f(x) = 4 - x^2\) with \(x \geq 0\).

Step 5: Find Inverse Function

To find the inverse, solve \(y = 4 - x^2\) for \(x\):1. Swap \(x\) and \(y\): \(x = 4 - y^2\).2. Rearrange to solve for \(y\): \(y^2 = 4 - x\).3. Take the square root: \(y = \sqrt{4 - x}\).Thus, the inverse function is \(f^{-1}(x) = \sqrt{4 - x}\), where \(x \leq 4\).

Step 6: Verify Domain and Range of Inverse

The inverse function \(f^{-1}(x) = \sqrt{4 - x}\) is valid for the domain \(x \leq 4\), corresponding to the range \(y \geq 0\) from the original function's restricted domain.

Key Concepts

Quadratic Functions

Quadratic functions are a type of polynomial function represented by the general form \(f(x) = ax^2 + bx + c\). The key feature of a quadratic function is its U-shaped graph, called a parabola.

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards, like in our exercise \(f(x) = 4-x^2\).

The point where the parabola changes direction is called the vertex. In our exercise, the vertex is calculated at \((0, 4)\), which is the highest point of the parabola since it opens downwards. Understanding the direction and position of the vertex helps in analyzing the function's behavior and in performing domain restriction to make it one-to-one.

Domain Restriction

The domain of a function is the complete set of possible values of the independent variable \(x\). For quadratic functions like \(f(x) = 4-x^2\), if the entire real line is considered, it leads to a graph that is symmetric and not one-to-one.To make the function one-to-one, we need to "restrict" the domain. This involves choosing a part of the domain where the function is either strictly increasing or strictly decreasing.

• For the function \(f(x) = 4-x^2\) with a vertex at \((0, 4)\), we can choose a domain like \(x \geq 0\) where the function decreases as \(x\) increases.
• Alternatively, \(x \leq 0\) is another option, with the function increasing as \(x\) decreases.

By restricting the domain in this way, we maintain the function as one-to-one, which is critical when finding its inverse.

One-to-One Function

A one-to-one function is a type of function in which every element of its range is mapped to by exactly one element of its domain. When plotted, no horizontal line intersects the function's graph more than once. This property is essential for a function to have an inverse.For quadratic functions like in the exercise, the original function \(f(x) = 4-x^2\) is not one-to-one because its graph is a symmetrical parabola.

• By restricting the domain to either \(x \geq 0\) or \(x \leq 0\), the symmetry is removed, making the function one-to-one. This allows us to properly define an inverse function since each output from the function now corresponds to exactly one input.

Understanding one-to-one functions helps in managing transformations and applying mathematical operations correctly.

Function Inverse

The inverse of a function \(f\) is another function \(f^{-1}\) that reverses the actions of \(f\). For an inverse to exist, the function must be one-to-one. This is why domain restriction is so crucial.To find an inverse:

• Swap the variables \(x\) and \(y\) in the function equation: starting with \(y = f(x)\), rewrite it as \(x = f(y)\).
• Solve for \(y\) to get the inverse function \(f^{-1}(x)\).

For the restricted version of the function \(f(x) = 4 - x^2\) with \(x \geq 0\), the steps are as follows:1. Swap to \(x = 4 - y^2\).2. Rearrange to \(y^2 = 4 - x\).3. Solve for \(y\) by taking the square root to obtain \(y = \sqrt{4 - x}\).The result is the inverse function \(f^{-1}(x) = \sqrt{4 - x}\), valid for the domain \(x \leq 4\), reflecting the range of the original restricted function. Recognizing the inverse allows for reversing the input-output relationship accurately.