Problem 46 - Calculus of a Single Variable

Question

Find the inverse function of \(f,(\mathbf{b})\) graph \(f\) and \(f^{-1}\) on the same set of coordinate axes, ( \(\mathbf{c} )\) describe the relationship between the graphs, and ( \(\mathbf{d} )\) state the domain and range of \(f\) and \(f^{-1} .\) \(f(x)=\frac{x+2}{x}\)

Step-by-Step Solution

Answer

The inverse function of \(f(x)=\frac{x+2}{x}\) is \(f^{-1}(x)=\frac{2}{x-1}\). The graph of \(f^{-1}(x)\) is the reflection of the graph of \(f(x)\) about the line \(y=x\). The domain and range of both functions is (-∞, 0) ∪ (0, ∞).

Step 1: Find the Inverse Function

To find the inverse function, start by replacing \(f(x)\) with \(y\).\n So, \(y=\frac{x+2}{x}.\) Interchange \(x\) and \(y\) to find the inverse: \(x=\frac{y+2}{y}.\) Solve this equation for \(y\) to obtain the inverse function. Multiply both sides of the equation by \(y\) to get \(xy=y+2.\) Substract \(y\) from both sides to get \(xy-y=2.\) Factor out \(y\) to get \(y(x-1)=2.\) Finally, divide by \(x-1\) to get the inverse: \(f^{-1}(x)=\frac{2}{x-1}\). This is the inverse function of \(f(x)\).

Step 2: Graph \(f\) and \(f^{-1}\)

Plotting both \(f\) and \(f^{-1}\) on paper or online graphing tool will give proper visualization of the two functions. Locate the points on the graph carefully since we are dealing with fraction functions.

Step 3: Describe the Relationship

Looking at the obtained graphs, the curve for the inverse function should be a reflection of the function \(f(x)\) about the line \(y=x\). This relationship holds for all functions and their respective inverses.

Step 4: Determine the Domains and Ranges

The domain of a function is the set of all possible x-values and the range is the set of all possible y-values. For the original function \(f(x)=\frac{x+2}{x}\), x ≠ 0 because we cannot divide by zero. So, the domain of \(f\) is (-∞, 0) ∪ (0, ∞). By graphing \(f(x)\), we find that the range of \(f\) is also (-∞, 0) ∪ (0, ∞). For the inverse function \(f^{-1}(x)=\frac{2}{x-1}\), x ≠ 1 for the same reason. Therefore, the domain of \(f^{-1}\) is (-∞, 1) ∪ (1, ∞). Similarly, the range of \(f^{-1}\) is (-∞, 0) ∪ (0, ∞).

Key Concepts

Graphing Functions

Graphing functions is a fundamental aspect of understanding their behavior visually. For the function \( f(x) = \frac{x+2}{x} \), it is essential to recognize that the equation involves a rational expression. This means we should expect asymptotes, where the graph approaches a line but never actually touches it.

To graph \( f(x) \), it's helpful to identify key points and asymptotes:

• Vertical Asymptote: Occurs where the denominator equals zero. In this case, x = 0.
• Horizontal Asymptote: As \( x \to \infty \) or \( x \to -\infty \), the graph approaches y = 1.
• Intercepts: Find the y-intercept by evaluating \( f(0) \), but this is not applicable here as x cannot be zero. Check for x-intercepts as well.

Once these points are identified, draw the curve keeping in mind the nature of rational functions. Label the axes clearly and plot the function with its inverse for comparison.

Function Domain and Range

The domain and range of a function are crucial for understanding where the function is valid and what kind of outputs it can have. For the function \( f(x) = \frac{x+2}{x} \), it cannot take the value when x = 0, as division by zero is undefined.

This results in:

• Domain for \( f \): All real numbers except 0, expressed as (-∞, 0) ∪ (0, ∞).

For the inverse function \( f^{-1}(x) = \frac{2}{x-1} \), it is undefined at x = 1:

• Domain for \( f^{-1} \): All real numbers except 1, written as (-∞, 1) ∪ (1, ∞).

The range involves the possible outputs for the functions:

• Range for both functions: As seen from the graphs and equations, they can take any real value except 0. Thus, their range is (-∞, 0) ∪ (0, ∞).

This means the functions can produce positive and negative values but not zero.

Reflection Over Line y=x

An important relationship between a function and its inverse is the reflection over the line \( y = x \). This line serves as a mirror line, showing the symmetry of the inverse functions in relation to the original function.

When you graph \( f(x) = \frac{x+2}{x} \) and its inverse \( f^{-1}(x) = \frac{2}{x-1} \), observe that each is a reflection of the other across this line. For every point \((a, b)\) on the graph of \( f(x) \), there is a corresponding point \((b, a)\) on \( f^{-1}(x) \).

This specific symmetry helps verify that these functions are indeed inverses of one another. Visualizing it on a graph provides a powerful understanding of how functions and their inverses are interconnected, emphasizing the beauty and logic of function transformations.