Problem 72 - A Graphical Approach to College Algebra

Question

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\sqrt{x-8}, x \geq 8$$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = x^2 + 8\).

Step 1: Understand the Definition of Inverse Functions

When we are asked to find the inverse of a function \(f\), we are essentially looking for a function \(f^{-1}\) such that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). This means that the inverse function reverses the effect of the original function.

Step 2: Swap and Solve for x

To find the inverse, start by swapping \(f(x)\) and \(x\) in the equation. So, we change \(y = f(x) = \sqrt{x-8}\) to \(x = \sqrt{y - 8}\). Now, solve for \(y\) to express \(y\) in terms of \(x\).

Step 3: Square Both Sides to Remove the Square Root

Square both sides of the equation \(x = \sqrt{y - 8}\) to eliminate the square root: \(x^2 = y - 8\).

Step 4: Solve for y

Add 8 to both sides of the equation \(x^2 = y - 8\) to isolate \(y\): \(y = x^2 + 8\).

Step 5: Write the Inverse Function

Now that we have expressed \(y\) in terms of \(x\), the inverse function is \(f^{-1}(x) = x^2 + 8\).

Key Concepts

One-to-One Functions: What and Why?

A one-to-one function is a special type of function that really helps when you need to find an inverse function. If a function is one-to-one, it means each output value is paired with one unique input value. Much like matching each person at a party to a unique seat, no two people can sit in the same seat, making the allocation neat and orderly.
This property is crucial. Why? Because it ensures that when we do reverse calculations, we land back where we started. Without the one-to-one nature of a function, we might have multiple input values pointing to the same output. Consequently, when trying to reverse or find an inverse, it'd be hard to tell which starting point we came from!

This one-to-one trait is mathematically defined by stating that if \(f(a) = f(b)\) then \(a=b\). This assures that for every output, there's exactly one input. Verifying a function's one-to-one status can be done using a few methods:

• Graphically by checking if the function passes the Horizontal Line Test.
• Analytically by confirming if \(f(a) = f(b)\) implies that \(a=b\).

Mastering Algebraic Manipulation

Algebraic manipulation is like gymnastics for numbers; it allows us to twist and bend equations into forms or points of view that suit our needs. This skill is invaluable when finding inverse functions. By flipping and turning the original equation, we can reveal the inverse.
For this function, observe the steps: we began with the equation from the function and simply swapped the knowns and unknowns, changing from \(y = f(x) = \sqrt{x-8}\) to \(x = \sqrt{y-8}\). This swap kicks off our journey to unearth the inverse function.
Next up, the task was removing the square root to bring our equation into a more usable form. By squaring both sides, \(x^2 = y-8\) cleared the hurdle of the square root. This process of squaring is aimed at simplifying operations so that we can isolate our variable of interest, \(y\), on one side by further algebraic maneuvers. In our example, adding 8 finalized the journey: \(y = x^2 + 8\). The beauty of these manipulations is they help to transform equations into easy, recognizable forms to solve for desired variables.

Understanding Function Transformations

With function transformations, we learn how changes in equations manifest visually as shifts, stretches, or flips in their associated graphs. Consider our original function \(f(x)=\sqrt{x-8}\). Recognizing transformations can illuminate how to manage and inversely navigate these equations.

• The \(\sqrt{x-8}\) represents a shift to the right by 8 units compared to the basic square root function \(\sqrt{x}\). This horizontal shift is due to the \(x-8\) inside the square root, suggesting that every point is moved 8 units right on the x-axis.
• The inverse function \(f^{-1}(x) = x^2 + 8\) serves as a transformation mirror. It suggests a graph starting from the line \(y=x^2\) but moved upwards by 8 units, as indicated by the constant added outside the square term.

Understanding these transformations deepens comprehension of how changes in one form lead to systematic echoes or compensations in the other. Moreover, these insights lead to a deeper grasp of how one function’s template is reflected in its inverse, cementing the connection between an original function and its inverse.