Problem 33 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine algebraically whether the function is one-to-one. If it is, find its inverse function. Verify your answer graphically. $$f(x)=x^{2}+5, x \geq 0$$

Step-by-Step Solution

Answer

The function \(f(x)=x^{2}+5\), for \(x \geq 0\) is one-to-one because it consistently increases as x increases. The inverse of this function is \(f^{-1}(x) = \sqrt{x - 5}\) for \(x \geq 5\). The graphical representation of the function and its inverse verify the obtained results.

Step 1: Check for one-to-one function

A function is one-to-one if it has at most one output for each input. To test this, note that \(f(x)=x^{2}+5\) for \(x \geq 0\) increases as x increases. So, it is a one-to-one function.

Step 2: Finding the inverse

In order to find the inverse of the function, you can replace \(f(x)\) with \(y\), creating the equation \(y = x^{2}+ 5\). Then solve for x, yielding \(x = \sqrt{y-5}\). Because \(x \geq 0\), the square root of \(y - 5\) must also be greater than or equal to 0. The inverse function, \(f^{-1}(x)\), is \(f^{-1}(x) = \sqrt{x - 5}\) for \(x \geq 5\).

Step 3: Graphical verification

To graphically verify the result, plot both the original function and its inverse on the same graph. Use a set of values for x and calculate the corresponding y values for both the original function and its inverse. Note that the function and inverse should mirror each other about the line \(y=x\). The reflection of the graph \(y=x^{2}+5\), would result in the graph \(y=\sqrt{x-5}\), which is indeed what was found algebraically.

Key Concepts

Finding the Inverse Function

Understanding the concept of an inverse function is crucial when dealing with algebraic functions. An inverse function essentially reverses the operation of the original function. For the function in question, \[ f(x) = x^{2}+5, x \geq 0 \], we observed that it is one-to-one. This means there is a unique y-value for every x-value within the domain, which permits the existence of an inverse function.

The process of finding an inverse includes several steps. First, express the function as \( y = x^{2} + 5 \) and then interchange the roles of x and y, leading to the equation \( x = y^{2} + 5 \). To solve for y, we need to perform inverse operations, which in this case, involves taking the square root of both sides of the equation. However, one subtle point is ensuring we do not introduce any additional values that the original function didn't account for; since the domain of \( x \) is \( x \geq 0 \) or \( x \) is non-negative, we only consider the positive square root. Therefore, the inverse function is \( f^{-1}(x) = \sqrt{x - 5} \) with the domain \( x \geq 5 \) to maintain consistency with the original function.

Graphical Verification of Functions

Graphical representation can be a powerful tool to confirm our algebraic findings. The relationship between a function and its inverse is seen clearly when they are plotted on the same set of axes. Ideally, they reflect symmetrically over the line \( y = x \).

To verify our result for \( f^{-1}(x) = \sqrt{x - 5} \) graphically, we plot both the original function \( f(x) = x^{2}+5 \) and its inverse on a graph. As we plot points for each function and connect them, we observe that the graph of the inverse is indeed a reflection across the line \( y = x \). This mirrors the steps taken algebraically and provides a visual confirmation that the inverse function has been properly identified. When performing these plots, particularly with students new to the concept, using a graphing calculator or software can be very helpful, allowing them to see the symmetry between the function and its inverse intuitively.

Algebraic Functions Analysis

Analyzing algebraic functions is a fundamental skill, which includes verifying if the function is one-to-one and if an inverse function exists. A function is one-to-one if each element of the range is uniquely paired with an element of the domain. To determine this property, we look at the function's formula and graph, or apply the Horizontal Line Test.

In our example, the function \( f(x) = x^{2}+5 \) is one-to-one within the domain \( x \geq 0 \) due to its increasing nature. Mathematically, this means the derivative of the function is always positive, which is another test for one-to-oneness. Once we establish this, we look for the inverse by algebraically manipulating the function and ensuring that the resulting formula aligns with the original function's domain and range. The algebraic examination provides a structured approach to understanding the characteristics and behaviors of functions, which is essential for further mathematical studies.