Problem 547 - Precalculus

Question

Find the inverse of the function on the given domain. \(f(x)=(x-4)^{2}, \quad[4, \infty)\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Function Type

The given function is \(f(x)=(x-4)^{2}\), which is a quadratic function. Since it is defined on the domain \([4, \infty)\), it is a one-to-one function on this interval, allowing it to have an inverse.

Step 2: Replace \(f(x)\) with \(y\)

We start by rewriting the function \(f(x) = (x - 4)^2\) as \(y = (x - 4)^2\). This helps us set up the equation for finding the inverse.

Step 3: Solve for \(x\) in terms of \(y\)

To find the inverse, we need to express \(x\) in terms of \(y\). Begin by taking the square root of both sides, keeping in mind the domain restriction: \[\sqrt{y} = x - 4\]

Step 4: Solve for \(x\)

Add 4 to both sides to isolate \(x\): \[\sqrt{y} + 4 = x\]Thus, \(x = \sqrt{y} + 4\).

Step 5: Replace \(x\) with \(f^{-1}(y)\)

To express the inverse function, replace \(x\) with \(f^{-1}(y)\), giving us: \[\f^{-1}(y) = \sqrt{y} + 4\]

Step 6: Replace \(y\) with \(x\)

Since the input of the inverse function is typically written as \(x\), replace \(y\) with \(x\) to give the final inverse function: \[\f^{-1}(x) = \sqrt{x} + 4\]

Key Concepts

Quadratic Functions

A quadratic function is a fundamental concept in algebra. It's typically expressed in the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). Quadratic functions graph as parabolas, which are symmetrical U-shaped curves. The vertex of the parabola is a significant point that can serve as a minimum or maximum, depending on the parabola's orientation. In our given function \( f(x) = (x-4)^2 \), the parabola opens upwards since the coefficient of \( x^2 \) is positive. Notice that this function has been written in vertex form: \( f(x) = (x-h)^2 \) with \( h = 4 \). This shows the parabola is shifted to the right along the x-axis by 4 units. Quadratic functions are not one-to-one over their entire domain, but they can be restricted to become one-to-one, which is essential for finding inverses.

One-to-One Function

A one-to-one function is a function where each output value is paired with exactly one input value. This means that no horizontal line intersects the graph in more than one point. In mathematical terms, a function \( f \) is one-to-one if, whenever \( f(a) = f(b) \), it follows that \( a = b \). For a function to have an inverse, it must be one-to-one.
Since the general form of quadratic functions is not one-to-one across their entire domain, we need to apply restrictions to their domain to make them one-to-one, enabling us to find their inverses. In the case of our example, the quadratic function \( f(x) = (x-4)^2 \) becomes one-to-one on the restricted domain \([4, \infty)\). This restriction ensures that each \( y \)-value in the function's range corresponds to only one \( x \)-value.

Restricted Domain

A restricted domain is a limitation placed on the set of input values for a function. This is often necessary for quadratic functions to make them one-to-one. Without such a restriction, quadratic functions are not one-to-one due to their symmetric nature. The domain of a function determines its behavior and transformations.
In this scenario, our function's domain was adjusted to \([4, \infty)\) to ensure it is one-to-one. The choice of this restriction allows only the part of the parabola that is increasing to be considered. This means each input value leads to a unique output, satisfying the one-to-one condition, and consequently enabling the creation of an inverse function.

Solving Equations

Solving equations is a fundamental process in mathematics, where you determine the values of the variables that satisfy the given equation. In our example, solving the equation for the inverse of a restricted quadratic function involves several algebraic steps.
1. Start by expressing the quadratic function \( f(x) = (x-4)^2 \) as \( y = (x-4)^2 \). This sets up the equation you need for inversion. 2. Then, solve for \( x \) in terms of \( y \) by taking the square root of both sides. This must be done cautiously, considering the domain restrictions to maintain validity. You'll reach \( \sqrt{y} = x - 4 \).
3. Isolate \( x \) by adding 4 to both sides: \( x = \sqrt{y} + 4 \). 4. Finally, switch \( y \) to \( x \) to complete the expression for the inverse function \( f^{-1}(x) = \sqrt{x} + 4 \). Each step follows logically from mathematical rules, ensuring that the function's inverse reflects its behavior accurately on the restricted domain.