Problem 72 - A Graphical Approach to Precalculus with Limits

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: Determine the domain of the function

The function given is \( f(x) = \sqrt{x-8} \). For the square root function to be defined, the expression inside the square root must be non-negative. Thus, the domain of \( f(x) \) is \( x \geq 8 \).

Step 2: Express the function in terms of y

Replace \( f(x) \) with \( y \) to express the function in terms of \( y \): \[ y = \sqrt{x-8} \] We need to solve this equation for \( x \) to find the inverse function.

Step 3: Square both sides to eliminate the square root

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

Step 4: Solve for x

To solve for \( x \), add 8 to both sides of the equation: \[ x = y^2 + 8 \].

Step 5: Express the inverse function

Now that we have \( x = y^2 + 8 \), replace \( y \) with \( x \) to express the inverse function: \[ f^{-1}(x) = x^2 + 8 \].

Key Concepts

One-to-One Functions

A one-to-one function is a special type of function where each input has a unique output, and no two different inputs map to the same output. This means that if you take any two distinct inputs, the outputs for each will always be distinct as well. Such functions are also referred to as injective functions.
Understanding if a function is one-to-one is essential, especially when you want to find its inverse. Only one-to-one functions have inverses that are also functions.
To determine if a function is one-to-one, you can use the horizontal line test on its graph. If no horizontal line cuts the graph at more than one point, the function is one-to-one. Alternatively, in an algebraic approach, you can show that if \[ f(a) = f(b) \] leads to \[ a = b \], then the function is one-to-one. This implication confirms that different outputs are indeed produced by different inputs.

Square Root Functions

Square root functions involve using the square root of a variable. These functions are defined as \( f(x) = \sqrt{x} \) for a basic version, but can take forms like\( f(x) = \sqrt{x - a} \) where the operation is performed on \( x - a \).
These functions are peculiar because they only accept non-negative inputs. This characteristic comes from the property of square roots that necessitates non-negativity to remain within the realm of real numbers.
If you were to see a graph of a square root function, it would start at some point, often at \( x = a \), and rise progressively as \( x \) increases. This portion is called the principal square root, which avoids negative outputs.

• For \( f(x) = \sqrt{x-8} \), the function is defined only if\( x \geq 8 \).
• The graph starts at \( x = 8 \), capturing only positive values thereafter.

Ensure you recognize the initial point where the function is defined, which depends on what lies inside the square root.

Domain of a Function

The domain of a function is the complete set of possible values of the independent variable. Simply put, it is the set of all inputs for which the function is defined. Understanding the domain is crucial as it tells us the range of values for which we can apply the function rules.
As functions can be restricted by various means, it's essential to apply appropriate conditions to determine their domain. For instance, functions involving a square root have restrictions to ensure no negative numbers appear under the square root symbol.

• For \( f(x) = \sqrt{x-8} \), the expression inside the square root (\( x-8 \)) must be zero or positive to avoid imaginary numbers.
• This restriction gives us \( x \geq 8 \) as the domain for the function.

Remember, checking the domain is an initial and crucial step when dealing with functions. By recognizing these limitations, we ensure we apply the function correctly over its valid range of inputs.