Problem 35 - 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 \(f^{-1} .\) \(f(x)=2 x-3\)

Step-by-Step Solution

Answer

The inverse function of \(f(x)=2x-3\) is \(f^{-1}(x)=\frac{x+3}{2}\). The graphs of \(f(x)\) and \(f^{-1}(x)\) are reflections in the line \(y=x\). For both the functions \(f(x)\) and \(f^{-1}(x)\), the domain and range are all real numbers.

Step 1: Finding the Inverse function

To find the inverse of the function, replace \(f(x)\) with \(y\). This gives us the equation \(y = 2x - 3\). Next, we have to switch \(x\) and \(y\) to get \(x = 2y - 3\). Afterwards, solving for \(y\), gives the inverse function. \[y = \frac{x+3}{2}\] Hence, the inverse of the function \(f(x) = 2x - 3\) is \(f^{-1}(x) = \frac{x+3}{2}\).

Step 2: Graphing the functions

The graph of the original function \(f(x)=2x-3\), is a line with a slope of 2 and a y-intercept of -3. The inverse function \(f^{-1}(x)=\frac{x+3}{2}\) will be a line with a slope of 1/2 and a y-intercept of 3/2. The functions are graphed on the same set of axes, they are reflection of each other along the line \(y=x\).

Step 3: Describing the relationship

The functions \(f(x)\) and \(f^{-1}(x)\) are reflections of each other in the line \(y = x\). This means that if you take any point on the graph of \(f(x)\), reflect it in the line \(y = x\), you will obtain a corresponding point on the graph of \(f^{-1}(x)\).

Step 4: Domain and Range

The domain of a function is the set of all real numbers `x` for which the function is defined. The range of a function is the set of all real numbers `y` that the function can produce. For \(f(x) = 2x - 3\), because this is a linear function, its domain and range are all real numbers. The function \(f^{-1}(x)=\frac{x+3}{2}\) is also a linear function. Hence, its domain and range are also all real numbers.

Key Concepts

Linear Functions

Linear functions are fundamental in mathematics. They are expressed in the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants.
The variable \( x \) represents the input value, while the output is \( f(x) \). These functions create straight lines when graphed.
Here are some key features of linear functions:

• **Slope**: The slope \( a \) determines the steepness of the line. A larger slope means a steeper line.
• **Intercept**: The \( y \)-intercept \( b \) is the point where the line crosses the \( y \)-axis. It's what you get when setting \( x = 0 \).

Understanding these two aspects allows you to draw and interpret any linear function's graph.
For the function in our example, \( f(x) = 2x - 3 \):

• The slope \( a \) is 2, indicating a line rising sharply.
• The \( y \)-intercept is -3, crossing the \( y \)-axis at \( (0, -3) \).

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the relationship between input (\( x \)) and output (\( y \)). For a linear function, you'll see a straight line.
To graph the function \( f(x) = 2x - 3 \):

• Start by plotting the \( y \)-intercept: the point at \( (0, -3) \). This is where the line crosses the \( y \)-axis.
• Use the slope to find another point. Since the slope is 2, it means for every 1 unit increase in \( x \), \( y \) increases by 2 units.
• Plot another point using this rise over run: move 1 unit right and 2 units up from \( (0, -3) \) to \( (1, -1) \).
• Draw a straight line through your points to complete the graph of \( f(x) \).

When graphing the inverse function \( f^{-1}(x) = \frac{x+3}{2} \), follow similar steps. This function is essentially reflected over the line \( y = x \).
Graphing both functions on the same axes lets you visually confirm their relationship, seeing how they mirror each other across \( y = x \). This reflection is a key feature of inverse functions.

Domain and Range

The domain and range of a function deal with the possible values of \( x \) and \( y \), respectively.
This concept is crucial for understanding how a function behaves overall and where it can "live."

• **Domain**: The set of all possible input values (\( x \)) that a function can accept.
• **Range**: The set of all possible output values (\( y \)) a function can produce.

For linear functions like \( f(x) = 2x - 3 \), both the domain and the range are all real numbers. Why?

• **Domain**: Any real number can serve as an input because there's no restriction on \( x \).
• **Range**: As \( x \) varies over all real numbers, \( y \) can also take any real value. Hence, the range is all real numbers as well.

Similarly, for the inverse function \( f^{-1}(x) = \frac{x+3}{2} \), both the domain and range remain all real numbers. This consistent behavior is typical for inverse functions when dealing with linear functions. In conclusion, understanding domain and range helps to entirely delineate the boundaries of a function’s application in mathematics.