Problem 63 - Algebra 2

Question

Find the inverse of each function. Is the inverse a function? $$ f(x)=\frac{1}{5} x^{3} $$

Step-by-Step Solution

Answer

The inverse of the function \(f(x) = \frac{1}{5} x^3\) is \(f^{-1}(x) = \sqrt[3]{5x}\) and yes, the inverse is a function.

Step 1: Replace f(x) With y

Replace \(f(x)\) with \(y\) to get \(y=\frac{1}{5}x^3\).

Step 2: Swap x With y

The next step involves swapping \(x\) and \(y\). So the equation becomes \(x=\frac{1}{5}y^3\).

Step 3: Solve for y

In order to find the inverse, the equation needs to be solved for \(y\). This involves isolating \(y\) on one side. First, multiply both sides of the equation by 5 to get rid of the fraction: \(5x = y^3\). Then, take the cube root of both sides to isolate \(y\): \(y = \sqrt[3]{5x}\). This is the inverse of the original function.

Step 4: Check if Inverse is a Function

To verify if the inverse function \(y = \sqrt[3]{5x}\) is a function or not, apply the horizontal line test on the graph of the function. If any horizontal line cuts the graph at more than one point, then the function does not pass the test and thus the original function does not have an inverse that is also a function. Here, any horizontal line would cut the graph at only one point, hence the inverse is indeed a function.

Key Concepts

Cube Root

When discussing functions and their inverses, understanding the concept of a cube root becomes essential. A cube root essentially "undoes" the operation of cubing a number. If you have a number, say let it be notated as \( a \), and you cube it to obtain \( b \), then cube root of \( b \) will give you back \( a \). Mathematically, this relationship is shown as \( a = \sqrt[3]{b} \). This process is quite similar to how square roots work, but it takes the three-dimensional volume into consideration.

In the context of our exercise, we took the cube root in order to derive the inverse function. From the step involving \( y^3 = 5x \), taking the cube root of both sides was necessary to isolate \( y \): we ended up with \( y = \sqrt[3]{5x} \). This operation is one of the fundamental techniques in finding inverse functions, particularly where polynomial equations are involved.

Function Inverse

The idea of finding an inverse function is about mapping back from output values to their original input values in a function. If \( f(x) \) takes an \( x \) and produces \( y \), then the inverse function, expressed as \( f^{-1}(x) \), takes \( y \) and brings you back to \( x \). This is crucial for solving many real-world problems where you need to reverse an operation or process.

There are general steps to find the inverse of a function, as shown in the exercise. First, switch the dependent and independent variables; in our case, we swapped \( x \) and \( y \) from the equation. This gives us a new equation, which we solve for \( y \). The solved \( y \) represents \( f^{-1}(x) \), our inverse function. For our exercise, the inverse of \( f(x) = \frac{1}{5}x^3 \) was found to be \( f^{-1}(x) = \sqrt[3]{5x} \). Remember, finding an inverse function is synonymous with reversing a function's effect.

Horizontal Line Test

To determine if an inverse is also a function, we utilize the horizontal line test on the graph of the original function. This test examines whether any horizontal line will intersect the graph of the original function more than once. If it does, the function is not one-to-one and thus does not have an inverse that is also a function.

In simpler terms, a function must be one-to-one to have an inverse that is also a function. For our exercise: \( f(x) = \frac{1}{5}x^3 \) passed the horizontal line test. Since any horizontal line will cut the graph at most once, it's confirmed that the inverse \( f^{-1}(x) = \sqrt[3]{5x} \) is indeed a function. This step ensures that returning to the original input values through the inverse is consistent and unique.

Understanding whether the inverse is a function is particularly helpful in various fields, from mathematics to engineering, as it guarantees reliable reverse mapping of data points.