Question

Determine whether the inverse of \(f\) is a function. Then find the inverse. \(f(x)=\frac{3}{x^2}+1\)

Short answer

The inverse of the given function is indeed a function. Its mathematical expression is \(f^{-1}(x) = \sqrt{\frac{3}{x-1}}\) and its domain is \(x>1\).

Step-by-step solution

Applying the Horizontal Line Test

The graph of our given function \(f(x)=\frac{3}{x^2}+1\) is represented by a vertical parabola which doesn't intersect any horizontal line more than once. So, the inverse of this function is itself a function.

Switch Function to a 'y' notation

Replace \(f(x)\) by \(y\). So, our equation becomes \(y=\frac{3}{x^2}+1\)

Switch 'x' and 'y'

Now, switch 'x' and 'y' to get: \(x=\frac{3}{y^2}+1\)

Solve for 'y'

Solve the equation for 'y' to get the inverse. First, subtract '1' to get \(x-1=\frac{3}{y^2}\). Now, reciprocate the equation to get \(\frac{1}{x-1}=\frac{y^2}{3}\). Then take the square root of both sides to get \(y = \sqrt{\frac{3}{x-1}}\) which is the inverse of \(f(x) = \frac{3}{x^2}+1\). The domain has to be \(x>1\) for the inverse function to be real.

Key concepts

Horizontal Line Test

The Horizontal Line Test is a visual way to determine if a function has an inverse that is also a function. Imagine drawing horizontal lines through the graph of the function you're examining. If any of these lines intersect the graph at more than one point, the function does not pass the test.
For a function to have an inverse that is also a function, each horizontal line should intersect the graph at most once. This indicates that for every output or 'y' value, there is only one corresponding 'x' value.

• If a function passes the Horizontal Line Test, it is one-to-one. This allows the function to have an inverse.
• If it fails the test, the inverse will not be a function.

In our exercise, the function given was a parabola-shaped graph, vertically opened. It's important to note that this type of graph can vary greatly depending on the function's specifics. However, in this case, the graph of our specific function doesn't intersect any horizontal line more than once. Thus, it passes the Horizontal Line Test, confirming that the inverse is indeed a function.

Solving Equations

To find the inverse of a function, solving equations accurately is key. After identifying the function, usually represented in terms of 'y', the next step is to swap the roles of 'x' and 'y'. This transformation helps in finding the inverse.
For our given function, we first express it as:

• Replace the function notation with 'y', making the equation easier to handle, like a regular algebraic expression.
• We then swapped 'x' and 'y'. Now, it’s time to solve for 'y'.

The goal is to isolate the 'y' on one side of the equation. The process includes basic algebraic steps such as subtracting, reciprocating, and taking square roots:

• Begin by subtracting values to clear out unnecessary terms.
• Use reciprocation: this helps in simplifying fractions on both sides of the equation.
• Finally, solve for 'y' using operations like square rooting, while ensuring all mathematical rules are respected.

Once these steps are completed correctly, you find the inverse function as it was done with the original problem.

Domain and Range

Understanding a function's domain and range is crucial when dealing with inverse functions. The domain refers to all possible 'x' values a function can take, while the range is all resulting 'y' values from applying these 'x' values.
When finding the inverse of a function, the domain and range swap roles:

• The domain of the original function becomes the range of the inverse.
• The range of the original function becomes the domain of the inverse.

For the inverse we calculated, it was important to identify the domain where the function remains real and defined. In the context of the original problem, the inverse function’s domain requires that the value of 'x' must be greater than 1 to yield real results. Knowing these ranges helps in avoiding undefined behaviors, like division by zero or taking square roots of negative numbers, which ensures the function stays within a safe mathematical boundary.