Problem 64 - 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 \(y=x\). \(g(x)=x^{2}+1, \quad x \geq 0\)

Step-by-Step Solution

Answer

(a) The inverse is \(f^{-1}(x) = \sqrt{x - 1}\). (b) Both graphs reflect over the line \(y = x\).

Step 1: Understand the Given Function

The function given is \(g(x) = x^2 + 1\) with the condition \(x \geq 0\). This condition makes the function one-to-one, as it restricts \(x\) to non-negative values, making sure it does not repeat any \(y\) values in the range.

Step 2: Set Up to Find the Inverse

To find the inverse of a function, replace \(g(x)\) with \(y\). So, the equation becomes \(y = x^2 + 1\). We will then solve for \(x\) in terms of \(y\).

Step 3: Solve for x

Subtract 1 from both sides to get \(y - 1 = x^2\). Then take the square root of both sides to solve for \(x\): \(x = \sqrt{y - 1}\).

Step 4: Write the Inverse Function

The inverse function swaps \(x\) and \(y\), so its form is \(f^{-1}(x) = \sqrt{x - 1}\). This is only valid for \(x \geq 1\) because the square root function is defined for non-negative values only.

Step 5: Graph the Function and Its Inverse

Plot the original function \(y = x^2 + 1\) for \(x \geq 0\), which will start at \(y = 1\) when \(x = 0\) and rise upwards. Plot the inverse function \(y = \sqrt{x - 1}\) for \(x \geq 1\), which will start at \(x = 1\) when \(y = 0\) and rise as \(x\) increases. Both should be reflected over the line \(y = x\).

Step 6: Verify Reflection Over y = x

Ensure that each point on \(g(x)\) has a corresponding point on the inverse function such that if \((a, b)\) is on \(g(x)\), then \((b, a)\) is on \(f^{-1}(x)\), demonstrating that these are reflections over the line \(y = x\).

Key Concepts

One-to-One Function

A one-to-one function is a special type of function where each input is paired with a unique output. In simpler terms, no two different inputs can give the same output. This property is crucial for a function to have an inverse that is also a function. So how can you tell if a function is one-to-one?
You can use the *Horizontal Line Test* on its graph. If any horizontal line crosses the graph at more than one point, then the function is not one-to-one.
For instance, consider the function \(g(x) = x^2 + 1\) when \(x \geq 0\). By restricting \(x\) to non-negative values, the function becomes one-to-one. This is because each value of \(x\) in this range results in a unique output, making it possible to find an inverse.

Graphing Functions

Graphing functions help visualize how inputs are transformed into outputs. To graph a function, you typically choose several values of \(x\), calculate their corresponding \(y\), and plot these points.
For the given function \(g(x) = x^2 + 1\), start graphing at \((0, 1)\), and as \(x\) increases, plot the points that follow the parabolic shape. This graph will open upwards, because it is a quadratic equation with a positive coefficient
Now, for the inverse function \(f^{-1}(x) = \sqrt{x - 1}\), graph it starting at \((1, 0)\) and, as \(x\) increases, trace the curve upwards.

• The original function starts at \((0, 1)\) and rises along the parabola.
• The inverse function starts at \((1, 0)\) and rises more steeply.

Graphing both of these together helps in visualizing how each function is the inverse of the other by comparing their respective curves.

Reflection Over Line y=x

The concept of reflecting a function over the line \(y = x\) demonstrates the relationship between a function and its inverse. This line acts as a mirror, and for a function's graph to reflect over it means that each point \((a, b)\) on the original function corresponds to a point \((b, a)\) on its inverse.
When you graph \(g(x) = x^2 + 1\) and its inverse, \(f^{-1}(x) = \sqrt{x - 1}\), alongside the line \(y=x\), you will observe the symmetry of their reflections:

• Each point on \(g(x)\) is a mirror image of a point on \(f^{-1}(x)\).
• If one point on \(g(x)\) is \((2, 5)\), the inverse point would be \((5, 2)\) on \(f^{-1}(x)\).

This symmetry visually confirms that you have correctly found the inverse. By understanding this reflection, students can better grasp how inverses work and why they're essential in mathematics.