Question

Determine whether the inverse of \(f\) is a function. Then find the inverse. $$f(x)=\frac{1}{x^2}+4$$

Short answer

The function \(f(x) = \frac{1}{x^2} + 4\) is not injective. Therefore, it does not have an inverse.

Step-by-step solution

Understand the task

We are given a function \(f(x) = \frac{1}{x^2} + 4\). We need to find out whether the inverse of this function exists and then calculate the inverse.

Check for injectiveness

By observing the function and plotting the graph, we realize that it is not injective since for \(x_1 \neq x_2\) we have \(f(x_1) = f(x_2)\). This is because squaring the inverse of \(x\) results in the same values for the positive and negative values of \(x\). For example, \(f(-2) = f(2)\). Thus, the function \(f(x)\) does not have an inverse.

Trying to find the inverse and conclusion

Although it was determined that the function \(f(x)\) is not injective and hence does not have an inverse, let's try to calculate the inverse anyway for the sake of the exercise. To find the inverse, we typically swap \(x\) and \(y\) and solve for \(y\). But since this function is not injective, its inverse won't be a function, and we won't proceed with this calculation. To conclude, the inverse of the function \(f(x) = \frac{1}{x^2} + 4\) does not exist.

Key concepts

Injective Function

An injective function, or one-to-one function, is a type of function where each element of the function's domain maps to a unique element in the codomain. This means that no two different inputs can produce the same output.

• For a function to be injective: if \( f(x_1) = f(x_2) \), then \( x_1 = x_2 \).
• In simpler terms, each input should map to exactly one output without any duplication of values.

When a function is injective, it becomes possible to define an inverse. This is because an inverse function will map each element of the function's codomain back to a single element in the domain.
Since the given function, \( f(x) = \frac{1}{x^2} + 4 \), is not injective—different inputs can give the same output—it cannot have an inverse that is also a function.

Function Inverse

A function's inverse essentially reverses the role of the output and the input of the original function. Finding an inverse involves swapping the dependent and independent variables and solving for the new dependent variable. Not all functions have inverses.

• To find an inverse: swap \( x \) and \( y \) in the function and solve for \( y \).
• A valid inverse function implies that it passes the horizontal line test on the graph of the original function, indicating injectiveness.

In our exercise, the original function \( f(x) = \frac{1}{x^2} + 4 \) was analyzed for an inverse. However, due to its non-injective nature, it fails the criteria for having an inverse that is also a function.

Non-injective Functions

Non-injective functions do not map each unique input to unique outputs; instead, different inputs might yield the same result. This characteristic limits the possibility of having an inverse function.

• In non-injective functions, \( f(x_1) = f(x_2) \) for \( x_1 eq x_2 \) can occur.
• Thus, these functions fail the horizontal line test, which means an inverse that is a function does not exist.

For the function \( f(x) = \frac{1}{x^2} + 4 \), since both positive and negative values of \( x \) yield the same output value, it is non-injective. This results in the absence of an inverse function, clearly seen when \( f(-2) = f(2) \). Therefore, finding an inverse results in a relation rather than a function.