Problem 73 - Algebra and Trigonometry

Question

A one-to-one function is given. (a) Find the inverse of the function. (b) Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line \(y=x .\) $$ g(x)=\sqrt{x+3} $$

Step-by-Step Solution

Answer

The inverse is \(g^{-1}(x) = x^2 - 3\). Graphing confirms they are reflections over \(y=x\).

Step 1: Understanding the Properties of Inverse Functions

To find the inverse of a one-to-one function like \(g(x) = \sqrt{x+3}\), we need to ensure the function is indeed one-to-one. Since the square root is applied on a linear expression \(x+3\), this function is increasing and hence one-to-one.

Step 2: Swap and Solve for x

For the inverse, we switch the roles of \(x\) and \(y\). Start with the equation \(y = \sqrt{x+3}\). Swap \(x\) and \(y\) to get \(x = \sqrt{y+3}\). Solve for \(y\) to find the inverse function.

Step 3: Solve for Inverse Form

Squaring both sides of the equation \(x = \sqrt{y+3}\) gives \(x^2 = y + 3\). To solve for \(y\), subtract 3 from both sides, resulting in \(y = x^2 - 3\). Thus, the inverse function is \(g^{-1}(x) = x^2 - 3\).

Step 4: Verify by Graphing

Graph the function \(g(x) = \sqrt{x+3}\) and its inverse \(g^{-1}(x) = x^2 - 3\) on the same set of axes. Additionally, plot the line \(y = x\). Verify that the function and its inverse are reflections over the line \(y = x\), confirming they are inverse functions.

Key Concepts

One-to-One Functions

A one-to-one function is a special type of function where each input is paired with a unique output. In other words, no two different inputs produce the same output. This property is essential when determining if a function has an inverse that is also a function. To recognize a one-to-one function, you can use the *horizontal line test*. Visually, if any horizontal line intersects the graph of the function more than once, the function is not one-to-one. For instance, the function given in the exercise, \(g(x) = \sqrt{x+3}\), passes this test.- **Key Property**: Only one-to-one functions have inverses that are also functions. - **Example**: Linear functions like \(f(x) = 2x + 1\) are one-to-one if their slope is not zero, because they continuously grow or decrease without repeating output values.

Function Graphing

Graphing functions and their inverses is a crucial skill that helps visualize their relationship. When you graph a function and its inverse, the graphs should appear as mirror images across the line \(y=x\).Here’s how you can graph a function and its inverse:1. **Plot the Original Function**: Start by graphing the original function, like \(g(x) = \sqrt{x+3}\). This represents the relationship between the input \(x\) and the output \(y\).
2. **Calculate the Inverse**: For the inverse function \(g^{-1}(x) = x^2 - 3\), plot its graph on the same set of axes.
3. **Draw the Line \(y = x\)**: This is the line over which both graphs should reflect, acting as a sort of "mirror".After graphing, observe if the function and its inverse are symmetric about the line \(y=x\). This symmetry confirms that the inverse calculation was done correctly and that both graphs correspond to inverse functions.

Properties of Inverse Functions

Inverse functions reverse the effect of the original functions. They take the output of the original function back to the input value. Knowing the properties of inverse functions is crucial for verifying and understanding them.Some important properties include:- **Reflection Property**: As noted earlier, the graph of a function and its inverse are mirror images across the line \(y=x\). This reflects the idea that inverse functions revert each other's operations.- **Operational Interchange**: For the function \(f\) and its inverse \(f^{-1}\), it holds that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). This means applying the function followed by its inverse (or vice versa) will get you back to your original value. Understanding these properties helps validate whether a function's derived inverse is accurate and operationally sound. In the exercise example, using these concepts, we've verified the inverse function that \(g^{-1}(x) = x^2 - 3\). This confirms that \(g\) and \(g^{-1}\) are indeed valid inverses.