Problem 36 - Algebra 2

Question

Find the inverse of each function. Is the inverse a function? $$ f(x)=\frac{3 x^{2}}{4} $$

Step-by-Step Solution

Answer

The inverse of the function \( f(x)=\frac{3 x^{2}}{4} \) is \( y = \pm \sqrt{\frac{4x}{3}} \), and it is not a function because it doesn't pass the vertical line test.

Step 1: Interchange x and y in the equation

Swap the roles of 'y' and 'x'. This means we replace every 'x' with 'y' and 'y' with 'x' in the function equation, resulting in \( x=\frac{3 y^{2}}{4} \). This step gets us closer to finding the inverse by putting our original function into a form that allows us to easily solve for 'y'.

Step 2: Solve for y

Rearrange the equation \( x=\frac{3 y^{2}}{4} \) to solve for 'y': Multiply each side by 4 to get \( 4x = 3y^{2} \), and then divide each side by 3, resulting in \( y^{2} = \frac{4x}{3} \). The final step to solve for 'y' is to take the square root of each side, giving us \( y = \pm \sqrt{\frac{4x}{3}} \). The resulting equation \( y = \pm \sqrt{\frac{4x}{3}} \) is the inverse of the original function.

Step 3: Determine if the inverse is a function

The inverse \( y = \pm \sqrt{\frac{4x}{3}} \) is not a function because it violates the vertical line test. The vertical line test states that any vertical line drawn through the function will intersect the function at most once. In this case, it is possible to draw a vertical line that intersects the inverse function more than once, meaning that it fails the vertical line test. Therefore, the inverse is not a function.

Key Concepts

Understanding Function Mapping

Function mapping is a fundamental concept in mathematics, where a function assigns each input exactly one output. This relationship between input and output is often described using the term "mapping." In the context of inverses, function mapping shows us how an input from the inverse needs to trace back to a single output in the original function's domain. In our example with the function \( f(x) = \frac{3x^2}{4} \), the function maps each value of \( x \) to a specific \( y \) value. To find the inverse, we reverse this mapping process. We swap 'x' and 'y' in the equation, giving us a new expression that represents the inverse relationship. Reversing this mapping isn't always as straightforward as it seems and can sometimes lead to outputs that don't satisfy the requirements of a function, which brings us to the concept of the vertical line test.

The Vertical Line Test in Practice

The vertical line test is a simple way to determine if a curve is the graph of a function. For any function, a vertical line should intersect its graph in at most one point. If this test fails, the relationship is not a function. In the case of our inverse, \( y = \pm \sqrt{\frac{4x}{3}} \), employing the vertical line test reveals something important. Because you can draw a vertical line that crosses the curve at two points simultaneously (due to the \( \pm \) sign), this relationship fails the test. This indicates that the inverse \( y \) value is not uniquely defined for every \( x \), and hence the inverse is not a function. This result helps students understand how the inverse of a given function might fail to qualify as a function itself due to this vital one-to-one mapping issue.

Strategies for Solving Equations

Solving equations is a critical skill to master in mathematics, especially when working with inverse functions. In this particular case, solving for the inverse function involved a few straightforward mathematical manipulations.First, we altered the original equation by swapping 'x' and 'y' to start unraveling the inverse. Then, solving the equation \( x = \frac{3y^2}{4} \) for 'y' required two key steps:

• Multiply both sides by 4 to eliminate the fraction, simplifying the equation to \( 4x = 3y^2 \).
• Divide by 3 to isolate \( y^2 \), leading to \( y^2 = \frac{4x}{3} \).

The final step was taking the square root, which introduced a \( \pm \) sign, resulting in \( y = \pm \sqrt{\frac{4x}{3}} \). These operations demonstrate how each step methodically peels back layers of the equation to reveal the inverse. Understanding the way math operations impact both sides of an equation is essential for effectively solving for inverse functions and finding solutions in general.