Problem 68 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine algebraically whether the function is one-to-one. Verify your answer graphically. If the function is one-to-one, find its inverse. $$f(x)=\sqrt{x-2}$$

Step-by-Step Solution

Answer

The function \(f(x)=\sqrt{x-2}\) is one-to-one, which is verified both algebraically and graphically. The inverse of this function is \(f^{-1}(x) = x^2 + 2\).

Step 1: Algebraic determination

A function is one-to-one, if for every \(x\), there corresponds at most one \(y\), and for every \(y\), there corresponds at most one \(x\). In simpler terms, each \(x\) goes to one \(y\), and each \(y\) comes from one \(x\). If we take \(f(x)=\sqrt{x-2}\), for every \(x\), there can only ever be one \(y\). Because there's only one option for square root (in this case real square root), there's no chance of getting two different \(y\)'s from one \(x\). Therefore, the function is one-to-one algebraically.

Step 2: Graphical verification

To graph \(f(x)=\sqrt{x-2}\), plot points considering domain starting from 2 (as square root of negative numbers are not real). You will find that the graph goes upward and to the right, but doesn't return down again or turn left. That means the graph passes the horizontal line test, i.e, a horizontal line will intersect the graph at most at one point, indicating that the function is one-to-one.

Step 3: Find the inverse

To find the inverse of the function, replace \(f(x)\) with \(y\). So, \(y=\sqrt{x-2}\). Now, swap \(x\) and \(y\): \(x=\sqrt{y-2}\). Then, square both sides and solve for \(y\) to get inverse function. It gives \((x)^2 = y-2\). Hence, the inverse function is \(f^{-1}(x) = x^2 + 2\).

Key Concepts

Inverse Functions

Understanding inverse functions is crucial when dealing with one-to-one functions. An inverse function, denoted as f^-1(x), basically swaps the roles of the input and output of the original function. To find an inverse, we first express the function in terms of y, i.e., y = f(x), and then switch x and y to get the equation of the inverse.

For our given function, f(x) = \(\sqrt{x-2}\), we find its inverse by replacing f(x) with y, yielding y = \(\sqrt{x-2}\). Then, we interchange x and y to obtain x = \(\sqrt{y-2}\) and proceed to isolate y. By squaring both sides, we eliminate the square root, leading to the equation x² = y - 2. After rearranging, we find the inverse f^-1(x) = x² + 2.

It's essential to remember that the inverse function undoes what the original function does, taking the output back to the original input. This property is evident when one compiles f(f^-1(x)) or f^-1(f(x)); in both cases, the result is x, demonstrating the inverse relationship.

Graphical Verification of Functions

Graphical verification is a visual method to confirm whether a function is one-to-one. To do this, we use the horizontal line test. If any horizontal line crosses the graph of the function at no more than one point, the function is one-to-one.

In the case of the function f(x) = \(\sqrt{x-2}\), we start graphing at the point where the function is defined, starting at x = 2 because the square root of negative numbers isn't real. As we plot the graph, it climbs upwards and rightwards without turning back or looping, which is the crucial part of passing the horizontal line test. This graphical behaviour confirms that there is a unique y for every x, reaffirming that f(x) is indeed a one-to-one function.

If a student were to plot this function, it would resemble half of an upward-opening parabola, shifted to the right by 2 units. The distinctive shape of the graph facilitates the graphical verification of the function's one-to-one characteristic.

Algebraic Determination of Function Characteristics

Apart from graphical methods, algebraic techniques play a fundamental role in determining the characteristics of a function. For a function to be one-to-one, every x in the domain must be paired with a unique y in the range, and vice versa.

In algebraic terms, this implies that if you select any two different x-values, say a and b, and if f(a) = f(b), then a must equal b. This condition, referred to as the definition of a one-to-one function, is rigorously used in algebraic determination of function characteristics.

For the given function f(x) = \(\sqrt{x-2}\), one-to-oneness can be algebraically verified by observing that the square root function is inherently a function that assigns one and only one output for every input. This means f(x) is injective, or one-to-one. Such algebraic insight complements the graphical verification and reinforces our understanding of the function's behavior over its domain.