Problem 78 - Precalculus : Building Concepts and Connections

Question

Two students have an argument. One says that the inverse of the function \(f\) given by the expression \(f(x)=6\) is the function \(g\) given by the expression \(g(x)=\frac{1}{6} ;\) the other claims that \(f\) has no inverse. Who is correct and why?

Step-by-Step Solution

Answer

The student who claimed that \(f(x) = 6\) has no inverse is correct. This is because neither of the conditions for a function and its inverse (\(f(g(x)) = x\) and \(g(f(x)) = x\)) were met when checking if \(g(x) = \frac{1}{6}\) was an inverse of \(f(x) = 6\). Furthermore, a constant function does not have an inverse in the set of all functions, as an inverse would have to map a single input to many outputs, which goes against the definition of a function.

Step 1: Recall definition of an inverse function

An inverse function \(g\) to a function \(f\) is a function that 'reverses' the operations of \(f\). Formally, if \(g\) is an inverse of \(f\), then \(f(g(x)) = x\) and \(g(f(x)) = x\). We'll use this definition to check whether \(g(x) = \frac{1}{6}\) is an inverse function to \(f(x) = 6\).

Step 2: Check if \(f(g(x)) = x\)

Apply \(f\) to \(g(x)\). In this case, wherever we see \(x\) in \(f(x)\), we substitute it with \(g(x) = \frac{1}{6}\). This gives us \(f(g(x)) = f(\frac{1}{6}) = 6\). This does not simplify to \(x\), Therefore, based on our condition for inverse functions, \(g\) can't be the inverse of \(f\), since we don't have \(f(g(x)) = x\).

Step 3: Check if \(g(f(x)) = x\)

Apply \(g\) to \(f(x)\). In this case, wherever we see \(x\) in \(g(x)\), we substitute it with \(f(x) = 6\). This gives us \(g(f(x)) = g(6) = \frac{1}{6}\). This does not simplify to \(x\), Therefore, based on our condition for inverse functions, \(g\) can't be the inverse of \(f\), since we don't have \(g(f(x)) = x\).

Step 4: Address the inverse of constant function

Finally, it is worth mentioning that a constant function like \(f(x) = 6\) does not have an inverse in the set of all functions. This is because constant functions map many inputs (all inputs, actually) to a single output. So, the inverse would have to map a single input to many outputs, which is not a function. From this, it is concluded that the second student, who claimed that \(f\) does not have an inverse, was correct.

Key Concepts

Constant Function

A constant function is a type of function where every input value maps to the same output value. In other words, no matter what value of 'x' you choose, the output will always be the same. For instance, the function expressed as \(f(x) = 6\) dictates that, irrespective of the input 'x', the output is consistently 6. This characteristic leads to a distinctive flat line when plotting the function on a graph.

A crucial point to understand about constant functions is that they do not pass the Horizontal Line Test, which checks if any horizontal line drawn across the graph intersects the function more than once. If it does, like it would for a constant function, this indicates that the function cannot have an inverse that is also a function. Constant functions deliver the same result for infinite inputs; therefore, establishing an inverse would mean one output linking back to countless inputs, defying the very definition of a mathematical function.

Overall, constant functions are simple yet pose unique limitations when considering inverses, as demonstrated in the exercise where the student correctly identified that a constant function does not possess an inverse.

Function Operations

Understanding function operations is vital in handling various algebraic expressions that involve more than one function. Some basic operations include addition, subtraction, multiplication, and division of functions. To perform these operations, you combine the corresponding outputs of the functions according to the operation in question. For example, if you have functions \(f(x)\) and \(g(x)\), their sum would be expressed as \((f + g)(x) = f(x) + g(x)\).

However, when it comes to finding the inverse function, we engage in a particular operation, we seek to 'reverse' the effect of a function. If we consider the function operation \(f(g(x))\), we’re computing the output of 'f' using the output of 'g' at 'x'. If 'g' is the inverse of 'f', then this operation should simply yield 'x'. The same goes for \(g(f(x))\), verifying that 'f' and 'g' are inverses.

Testing Inverses with Function Operations

When trying to determine if one function is the inverse of another, you apply each function to the other. If both \(f(g(x))\) and \(g(f(x))\) return the original input 'x', then 'f' and 'g' are indeed inverses. The students in the exercise utilized this method and discovered that the proposed function did not revert to the identity, thereby not fulfilling the criteria of being an inverse.

Definition of an Inverse Function

An inverse function, as the name suggests, reverses the action of a function. It is defined such that when one function takes an input 'x' and produces an output 'y', its inverse will take the output 'y' and return the original input 'x'. In a formal sense, if \(g\) is the inverse function of \(f\), then \(f(g(x)) = x\) and \(g(f(x)) = x\) for every 'x' in the domain of the original function.

For a function to have an inverse that is also a function, it must be bijective, meaning it must be both injective (one-to-one) and surjective (onto). Injective ensures that each output is linked to no more than one input, making the reverse mapping possible. Surjective makes sure that every possible output is achieved by some input, ensuring that the inverse has a complete set of outputs.

Unfortunately, not all functions have inverses, particularly when they are not one-to-one, which includes constant functions. This reflects the resolution of the debate presented in the exercise where one student contended that the inverse of \(f(x) = 6\) could not be \(g(x) = \frac{1}{6}\), as it would not satisfy the requirement that \(f(g(x))\) and \(g(f(x))\) both be 'x'. Their understanding of the definition of an inverse function was accurate, leading to the correct conclusion that the constant function \(f(x)=6\) has no inverse function.