Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the function on the given domain. $$ f(x)=12-x^{2},[0, \infty) $$

Step-by-Step Solution

Answer

The inverse is \(f^{-1}(y) = \sqrt{12 - y}\) for \(0 \leq y \leq 12\).

Step 1: Understand the Problem

We need to find the inverse of the function \( f(x) = 12 - x^2 \) for the domain \([0, \infty)\). Firstly, we must ensure that the function is one-to-one on the provided domain.

Step 2: Restrict Function Domain and Range

Since the function is defined for \([0, \infty)\), we check if the function is one-to-one. For \(x \geq 0\), \(12 - x^2\) decreases as \(x\) increases. Hence, it is one-to-one. The range of \(f(x)\) on this domain is \((-\infty, 12]\).

Step 3: Express \(y\) in terms of \(x\)

We start by letting \(y = f(x) = 12 - x^2\). Our goal is to solve for \(x\) in terms of \(y\), which will help us find the inverse function.

Step 4: Solve for \(x\)

Rearrange the equation \(y = 12 - x^2\) to find \(x\):\[ x^2 = 12 - y \] \[ x = \sqrt{12 - y} \]Since \(x\) must be non-negative, we only consider the positive square root.

Step 5: Write the Inverse Function

Thus, the inverse function is \(f^{-1}(y) = \sqrt{12 - y}\), for \(y \leq 12\) and \(y \geq 0\).

Step 6: Verify the Inverse

To verify, check if \(f(f^{-1}(y)) = y\) and \(f^{-1}(f(x)) = x\). For instance:- \(f(f^{-1}(y)) = 12 - (\sqrt{12-y})^2 = y\).- \(f^{-1}(f(x)) = \sqrt{12 - (12 - x^2)} = x\), which holds for \(x \geq 0\).

Key Concepts

One-to-One Function

A function is one-to-one if every output value is paired with exactly one unique input value. In simpler terms, no two different input values can produce the same output. This property is critical when we want to find the inverse of a function because only one-to-one functions have inverses that are also functions.
When assessing whether a function is one-to-one, you can use the Horizontal Line Test. If every horizontal line intersects the graph of the function at most once, then the function is one-to-one. For the function given in the exercise, \( f(x) = 12 - x^2 \), it is defined on the domain \([0, \infty)\). In this interval, the function consistently decreases, ensuring it is one-to-one.

• A continuously decreasing function on a restricted domain like \([0, \infty)\) maintains the one-to-one property.
• This ensures that each value in the range \((0, 12]\) corresponds to one unique \(x\) value.

Recognizing this characteristic allows us to proceed in finding its inverse function.

Function Domain and Range

The domain and range of a function provide crucial boundaries for understanding how the function behaves and where it exists. The domain consists of all possible input values, while the range includes all possible outputs.
For the function \( f(x) = 12 - x^2 \) given in our exercise, the domain is restricted to \([0, \infty)\). This restriction means we only consider non-negative \(x\) values. This is important because it excludes parts of the function where it would not be one-to-one.

• An essential step is to sometimes restrict the domain to achieve the one-to-one nature needed for inverses.
• For this quadratic function operating over \([0, \infty)\), the range is \((-\infty, 12]\), meaning the function produces values from slightly below 12 down to negative infinity as \(x\) increases within its domain.

Understanding these bounds helps clarify where the inverse function will exist. The inverse function's domain is the range of the original function and vice versa.

Solving Equations

To find the inverse of a function, you'll generally need to solve an equation by expressing the input in terms of the output. For the function \( f(x) = 12 - x^2 \), finding the inverse is achieved through a series of algebraic manipulations.
First, set the function equal to \(y\) to represent the output, then solve for \(x\):\[ y = 12 - x^2 \]Rearrange this to make \(x\) the subject:\[ x^2 = 12 - y \]\[ x = \sqrt{12 - y} \]Here are key points:

• We only consider the positive square root because the domain of \(x\) is non-negative \([0, \infty)\).
• The resulting inverse function is \( f^{-1}(y) = \sqrt{12 - y} \), where the inverse function's valid range is dictated by the original function's domain.

Lastly, verify the inverse by checking that plugging one function into the other returns the original value, ensuring the solution is correct.