Problem 189 - Calculus Volume 1

Question

For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function. $$ f(x)=x^{2}-4, x \geq 0 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt{x + 4} \) with domain \([-4, \infty)\) and range \([0, \infty)\).

Step 1: Understanding the Function

First, comprehend the given function \( f(x) = x^2 - 4 \) defined for \( x \geq 0 \). This is a quadratic function, but because of the restriction \( x \geq 0 \), it is one-to-one, making it possible to find its inverse.

Step 2: Find the Inverse Function

To find the inverse function, we start by replacing \( f(x) \) with \( y \):\[ y = x^2 - 4 \]Next, swap \( x \) and \( y \):\[ x = y^2 - 4 \]Solve for \( y \):\[ y^2 = x + 4 \]\[ y = \sqrt{x + 4} \]Since \( x \geq 0 \), \( y \) is non-negative, leading to the inverse function \( f^{-1}(x) = \sqrt{x + 4} \).

Step 3: Determine the Domain of the Inverse

The function \( f(x) = x^2 - 4 \) is defined for \( x \geq 0 \). Therefore, the output of the original function is \( y \geq -4 \). This implies that \( x+4 \geq 0 \) for \( f^{-1}(x) \), giving us \( x \geq -4 \). Therefore, the domain of the inverse function is \( [-4, \infty) \).

Step 4: Determine the Range of the Inverse

Because the domain of the original function \( f(x) \) is \( x \geq 0 \), the range of the inverse function \( f^{-1}(x) = \sqrt{x + 4} \) is all non-negative real numbers, i.e., \( [0, \infty) \).

Key Concepts

Domain of Inverse Function

When we talk about the domain of an inverse function, we're referring to the set of all possible input values the inverse function can accept. It's a crucial concept to grasp because it determines where our inverse function is defined and where we can use it.
In our exercise, we started with the function \( f(x) = x^2 - 4 \), defined for \( x \geq 0 \). The inverse function we found is \( f^{-1}(x) = \sqrt{x + 4} \).

Here’s how we determined the domain:

• First, understand that for any function \( g \), the domain of its inverse \( g^{-1} \) is the range of \( g \).
• For \( f(x) = x^2 - 4 \), the range is \( y \geq -4 \).
• Therefore, our inverse \( f^{-1}(x) \) is defined for \( x \geq -4 \), meaning its domain is \( [-4, \infty) \).

Understanding this helps us know where the inverse function \( f^{-1} \) makes sense, paving the way for solving further math problems.

Range of Inverse Function

The range of an inverse function tells us the set of all possible output values the function can produce. It's important because it tells us the capability and coverage of our inverse function.
The original function in our exercise \( f(x) = x^2 - 4 \) had its domain restricted to \( x \geq 0 \). When we found its inverse, \( f^{-1}(x) = \sqrt{x + 4} \), we then needed to find the range of this new function.

The steps we used include:

• The original function's domain \( x \geq 0 \) leads to a corresponding range for the inverse.
• The inverse function \( f^{-1}(x) = \sqrt{x + 4} \) produces non-negative values only.
• Thus, the range of \( f^{-1}(x) \) is \( [0, \infty) \), i.e., all non-negative real numbers.

Recognizing the range informs us about the scope of results we can achieve using \( f^{-1} \). This helps us understand not just the inverse function’s limits but also their importance in applications.

Quadratic Function

A quadratic function is a type of polynomial function with the general form \( ax^2 + bx + c \). It's important because it forms the basis for understanding parabolae in mathematics.
In the given exercise, our function \( f(x) = x^2 - 4 \) is a quadratic function simplified further because:

• The coefficient \( a \) is 1 (since the leading term is just \( x^2 \)).
• It doesn't have a \( bx \) term, making \( b = 0 \).
• The constant \( c \) is \(-4\).

Quadratic functions typically graph to form a curve called a parabola.

In this exercise, we have another interesting aspect: the restriction \( x \geq 0 \). This restriction makes the function one-to-one within that domain, which is key to finding an inverse.
Recognizing this restriction is essential, as not all quadratic functions have inverses without further conditions. Nevertheless, this mathematical property helps us process the function easily and understand its behavior more thoroughly.