Problem 123 - A Graphical Approach to Precalculus with Limits

Question

Using the given restrictions on the functions, find a formula for \(f^{-1}\). $$f(x)=-x^{2}+4, \quad x \geq 0$$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt{4 - x} \).

Step 1: Understanding Inverse Functions

To find the inverse function of a given function, we essentially switch the roles of dependent variable (output) and independent variable (input), and solve for the new output. The function we have is \( f(x) = -x^2 + 4 \), with the restriction \( x \geq 0 \). This restriction is important as it ensures that we can identify a unique inverse.

Step 2: Switch Variables

Start by replacing \( f(x) \) with \( y \) for simplicity: \( y = -x^2 + 4 \). To find the inverse, switch the roles of \( x \) and \( y \) to get \( x = -y^2 + 4 \). Now, our goal is to solve this equation for \( y \).

Step 3: Isolate the Quadratic Term

From the equation \( x = -y^2 + 4 \), isolate the quadratic term by subtracting 4 from both sides: \( x - 4 = -y^2 \). Next, multiply both sides by -1 to get \( y^2 = 4 - x \).

Step 4: Solve for the Inverse Function

Since \( x \geq 0 \) which was our initial condition, we only consider the positive square root to preserve this condition when solved. Therefore, we take the square root of both sides: \( y = \sqrt{4 - x} \). This gives us the inverse function \( f^{-1}(x) = \sqrt{4 - x} \).

Key Concepts

Quadratic Functions

Quadratic functions are a fundamental type of polynomial function that follow the general form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). These functions produce a parabolic graph, which is a smooth curve that opens upwards or downwards depending on the sign of \( a \). If \( a > 0 \), the parabola opens upward, while if \( a < 0 \), it opens downward.
\( f(x) = -x^2 + 4 \) is a specific example of a quadratic function, opening downwards due to the negative sign in front of the \( x^2 \) term. The vertex form of a quadratic \( y = a(x-h)^2 + k \) can be useful to easily determine the vertex of the parabola. In this case, the function reaches its maximum value at \( (0, 4) \). The restriction \( x \geq 0 \) indicates we only consider the right half of the parabola.
Quadratic functions are essential in understanding motion, optimization problems, and even financial calculations through their parabolic structures.

Function Restrictions

In mathematics, function restrictions are limitations or conditions placed on the domain and range of functions. They are vital in making sure functions provide meaningful results or match real-world situations accurately.
In the context of the exercise, the quadratic function \( f(x) = -x^2 + 4 \) has the restriction \( x \geq 0 \). This means we focus only on the values of \( x \) that are equal to or greater than zero, effectively considering only one "branch" of the quadratic's parabola. This is crucial to obtain a one-to-one function capable of having an inverse that uniquely associates each output with a single input.
Without such restrictions, quadratics being naturally not one-to-one would not have a valid inverse over their entire range. Such limitations are also seen in other mathematical contexts, such as square roots, where only non-negative inputs are valid in the standard real number system.

Square Root Function

Square root functions are specific types of functions that involve the square root of a variable. They are typically expressed as \( f(x) = \sqrt{x} \) and are defined for \( x \geq 0 \) because negative values do not yield real results when square-rooted in real numbers.
For the inverse derived in the exercise, \( f^{-1}(x) = \sqrt{4 - x} \), the square root function reflects the relationship from outputs back to their original inputs within the constrained range. This particular inverse retains its domain restriction: since the original function \( f(x) = -x^2 + 4 \) is defined only for \( x \geq 0 \), the inverse is also restricted, being valid only for \( x \leq 4 \) owing to the squared term within \( 4 - x \).
Understanding square root functions allows us to reverse operations effectively when paired with quadratics and is a core skill for tackling inverse problems.