Question

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

Short answer

The inverse of the function \(f(x)\) is indeed a function and is given by \(f^{-1}(x) = \frac{2}{x + 4}\)

Step-by-step solution

Make \(x\) the Subject

Start by making \(x\) the subject of the equation. This is achieved by first adding 4 to both sides of the equation and then taking the reciprocal, yielding \( x = \frac{2}{f(x) + 4} \)

Find the Inverse

The inverse of the function \(f(x)\) is sort by switching \(x\) and \(f(x)\). Hence, the inverse function, denoted by \(f^{-1}(x)\), becomes \(f^{-1}(x) = \frac{2}{x + 4}\)

Verify if Inverse is also a function

In this step, the inverse function undergoes the horizontal line test. For a function to have an inverse that is also a function, any horizontal line must intersect the graph at most once. Since the inverse function \(f^{-1}(x) = \frac{2}{x +4}\) is a rational function with no restrictions, we can tell that it passes the horizontal line test, as in every horizontal line, there is at most one intersection point with the graph of \(f^{-1}(x)\). Hence, the inverse is indeed a function.

Key concepts

Function Verification

Before jumping into finding the inverse of a function, it’s crucial to verify if the inverse itself is a function. The term "function verification" refers to this process. In simpler terms, after finding an inverse, you need to make sure that it behaves like a function does. So what exactly makes something a function? A function will have only one output for every single input. When verifying, our goal is to see if the inverse keeps this unique output rule intact.

The verification step is about double-checking. You ensure that no input value of your inverse maps to more than one output. For rational functions like the one in the exercise, this verification often rests on the behavior of the function graph and the equations themselves. Once you've confirmed an equation maps inputs to only unique outputs, you’ve successfully verified your function’s legitimacy.

Horizontal Line Test

The horizontal line test is an essential method to determine if a function has an inverse that is also a function. Imagine drawing horizontal lines across your graph.

• If any line crosses the graph more than once, the inverse will not be a function.
• If every horizontal line crosses at most once, your inverse is safe—meaning, it's a function too!

To apply this to the exercise, consider the inverse found: \(f^{-1}(x) = \frac{2}{x + 4}\). As you trace horizontal lines across this graph, no line ever intersects it more than once. Hence, the inverse must also be a function. This helps ensure inverse behavior remains consistent with function definitions.

Remember, the horizontal line test confirms you won't bump into duplicate outputs, thereby safeguarding the integrity of your function’s inverse. It's a visual check to keep functions on a reliable path when moving to and from inverse states.

Rational Functions

Rational functions are a vital part of understanding how various types work within function and inverse scenarios. A rational function is any function that can be expressed as the ratio of two polynomials. They often appear in the form \(\frac{p(x)}{q(x)}\), where both \(p(x)\) and \(q(x)\) are polynomials. This makes them quite versatile in mathematical tasks.

Our exercise involves the rational function \(f(x) = \frac{2}{x} - 4\). Its inverse turns into another rational function. Here, the primary complexity arises from ensuring the denominator never hits zero (since division by zero is undefined) and handling any asymptotic behavior.Rational functions like the one in the problem can exhibit some interesting behavior. They often have vertical asymptotes, where the graph shoots towards infinity, and horizontal asymptotes, underscoring a leveling off in graph behavior as \(x\) grows big.

• Vertical asymptotes occur where the denominator equals zero.
• Horizontal asymptotes reflect the eventual behavior as \(x\) heads to positive or negative infinity.

In the exercise context, these concepts help navigate through graph behavior and properties, ensuring solutions like inverse confirmations are accurately undertaken.