Question

In Exercises 35–46, determine whether the inverse of \(f\) is a function. Then find the inverse. $$ f(x)=x^4+2 $$

Short answer

So, the inverse of the given function is a function. The inverse of the function \(f(x)=x^4 + 2\) is \(f^{-1}(x)=\sqrt[4]{x-2}\).

Step-by-step solution

Rewrite the equation \(f(x)=y\)

Initially, rewrite the given function \(f(x)=x^4+2\) as \(y=x^4+2\). We're going to replace \(f(x)\) with \(y\) to simplify the problem.

Swap \(x\) and \(y\)

Swap \(x\) and \(y\) in this equation to obtain \(x=y^4+2\). This is a necessary step because we're trying to solve for the inverse. We switch \(x\) and \(y\) to achieve this.

Find the inverse function

To make \(y\) the subject, we subtract \(2\) from both sides of the equation to get \(x-2=y^4\). Taking the fourth root of both sides will give us the inverse function \(f^{-1}(x)=\sqrt[4]{x-2}\).

Check if the inverse is a function

We have to determine if this inverse passes the vertical line test to see if it is indeed a function. A vertical line crossing more than one point on the graph means that an input (\(x\)) has more than one output (\(y\)), which fails the definition of a function. However, the graph for \(\sqrt[4]{x-2}\) does pass the vertical line test, meaning every input (\(x\)) has only one output (\(y\)), so the inverse is indeed a function.

Key concepts

The Vertical Line Test

The vertical line test is an easy way to determine if a graph represents a function. Imagine drawing a vertical line, which is a straight line that goes up and down, on the graph of a mathematical equation.
If at any point, this line intersects the graph at more than one spot, then the equation does not represent a function. This is because, in a function, each input (or \(x\) value) should map to only one output (or \(y\) value).
For instance, if a vertical line crosses two points on a graph, it means a single \(x\) value is associated with two different \(y\) values. Thus, it fails the vertical line test. Conversely, if the vertical line touches only one point at any given \(x\), the graph successfully represents a function.
This test helps in verifying if the inverse function is valid or not. The graph of \(f^{-1}(x)=\sqrt[4]{x-2}\) passes this test since a vertical line will intersect any given \(x\) value at only one point.

Understanding Function Notation

Function notation is a way of representing equations in mathematics to easily manage functions and their behavior. In function notation, we typically express a function like \(f(x)\). Here, \(f\) represents the function itself, while \(x\) is the variable or input.
It is a convenient method for showing the dependency of one quantity on another. It makes it simple to convey operations and transformations applied to \(x\).

• The expression \(f(x)=x^4+2\) can be read as "\(f\) of \(x\)". This shows that the output is determined by raising \(x\) to the power of four, then adding 2.
• If we want to evaluate \(f\) at a specific input, say \(3\), we simply replace \(x\) with \(3\) to get \(f(3)=3^4+2\).

Using function notation allows us to clearly show operations on functions, like finding inverses, and makes them easier to manage and understand.

Solving Equations for Inverse Functions

Solving equations is a fundamental skill when dealing with inverse functions. To find the inverse of a given function, there are a few essential steps to follow. The goal is to "reverse" the operations performed by the original function on \(x\).

• First, express the function using \(y\), so for \(f(x)=x^4+2\), it becomes \(y=x^4+2\).
• Then, swap the variables \(x\) and \(y\). In this example, it becomes \(x=y^4+2\).
• The next step is to solve for \(y\) in terms of \(x\). Subtract \(2\) from both sides to get \(x-2=y^4\).
• Finally, take the fourth root to isolate \(y\), resulting in the inverse function, \[f^{-1}(x)=\sqrt[4]{x-2}\].

By following these steps, we convert the function back to a form that "undoes" the effect of \(f(x)\). Make sure to check its validity by confirming if the inverse function still holds the definition of a function with methods like the vertical line test.

Graphing Functions and Their Inverses

Graphing functions gives us a visual representation that can make mathematical concepts easier to understand. Plotting a function involves marking points based on the \(x\) inputs and their \(y\) outputs, then connecting these points smoothly.
When graphing an inverse function, the graph acts like a mirror reflection over the line \(y=x\). Here, the graph flips so that the original \(x\) values become \(y\) values and vice versa.

• The graph of the original function \(f(x)=x^4+2\) is a parabola opening upwards, starting from \(y=2\).
• For the inverse, \(f^{-1}(x)=\sqrt[4]{x-2}\), its graph starts at \(x=2\), since it can't accept values less than that due to the fourth root, and rises slowly.

Graphing both functions can confirm their relationship visually by checking that their graphs are symmetrical about the line \(y=x\). This reflective property is a key feature of inverse functions and visually reaffirms the computation done when solving for the inverse.