Question

The function \(f\) is one-to-one. (a) Find its inverse function \(f^{-1}\) and check your answer. (b) Find the domain and the range of \(f\) and \(f^{-1}\). (c) Graph \(f, f^{-1},\) and \(y=x\) on the same coordinate axes. $$ f(x)=-\frac{3}{x} $$

Short answer

Inverse function is \(f^{-1}(x) = -\frac{3}{x}\). Domain and range of both \(f\) and \(f^{-1}\) are all real numbers except 0.

Step-by-step solution

Write the function and express it

The given function is \[f(x) = -\frac{3}{x}\]. The goal is to find its inverse, denoted as \(f^{-1}(x)\).

Swap variables and solve for x

To find the inverse function, interchange the roles of \(x\) and \(f(x)\). Start with \[y = -\frac{3}{x}\].Switch \(x\) and \(y\) to get:\[x = -\frac{3}{y}\]. Now, solve for \(y\):\[y = -\frac{3}{x}\].

Express the inverse function

Since we have solved for \(y\), and \(y = f^{-1}(x)\), the inverse function is:\[f^{-1}(x) = -\frac{3}{x}\].

Check the inverse

To verify, we need to show that \[f(f^{-1}(x)) = x\] and \[f^{-1}(f(x)) = x\]. Start with \[f(f^{-1}(x))\]:\[f\left(-\frac{3}{x}\right) = -\frac{3}{-\frac{3}{x}} = x.\] Similarly, check \[f^{-1}(f(x))\]:\[f^{-1}\left(-\frac{3}{x}\right) = -\frac{3}{-\frac{3}{x}} = x.\] Both identities hold true.

Determine the domain and range

For the function \[f(x) = -\frac{3}{x}\], the domain is all real numbers except 0 (\(x eq 0\)). The range is also all real numbers except 0 because \(f(x)\) will never be zero. For the inverse function \(f^{-1}(x) = -\frac{3}{x}\), the domain is all real numbers except 0, and the range is also all real numbers except 0.

Graph the functions

To graph \(f(x)\), \(f^{-1}(x)\), and \(y = x\) on the same coordinate axes:- Graph \(y = -\frac{3}{x}\).- Graph its inverse \(y = -\frac{3}{x}\), which is the same in this case.- Plot the line \(y = x\) as a reference line that both functions are symmetric about.

Key concepts

headline of the respective core concept

An important topic in understanding the relationship between functions and their inverse is the concept of a one-to-one function. A function is said to be one-to-one if every element in the domain maps to a unique element in the range. This means that no two different inputs in the domain produce the same output in the range.

headline of the respective core concept

The terms domain and range are fundamental in the study of functions.
The domain of a function is the set of all possible input values (x-values) that the function can accept.
The range is the set of all possible output values (y-values) that the function can produce.
To find the domain of the function \( f(x) = -\frac{3}{x} \), we identify the values that x cannot take, in this case, x cannot be zero because division by zero is undefined. Therefore, the domain of f is all real numbers except zero.
Similarly, the range of the function f will also exclude zero for the same reason as explained above.
The domain and range of the inverse function \( f^{-1}(x) = -\frac{3}{x} \) will be the same, since the definition of the function and its inverse are the same.

headline of the respective core concept

A function graph offers a visual representation of the relationship between the input and output values of a function.
To graph a function, you typically start by plotting points based on pairs of input and output values.
In the case of the function \( f(x) = -\frac{3}{x} \), its graph will be a hyperbola that approaches but never touches the x-axis (where y would be zero) and the y-axis (where x would be zero). This is consistent with the domain and range exclusions of zero.
The inverse function \( f^{-1}(x) = -\frac{3}{x} \) will have the same graph because, as shown in the previous steps, the function and its inverse are identical.
Additionally, it's useful to graph the line \( y = x \) which serves as a reference line. Any point on the graph of the function f will have a corresponding point on the graph of the inverse function f^-1 that is symmetric about the line \( y = x \)