Problem 37
Question
a. Find the inverse of \(f(x)=x+1 .\) Graph \(f\) and its inverse together. Add the line \(y=x\) to your sketch, drawing it with dashes or dots for contrast. b. Find the inverse of \(f(x)=x+b(b \text { constant). How is the }\) graph of \(f^{-1}\) related to the graph of \(f ?\) c. What can you conclude about the inverses of functions whose graphs are lines parallel to the line \(y=x ?\)
Step-by-Step Solution
Verified Answer
The inverse of the function is \(f^{-1}(x) = x - 1\). Graphs of a function and its inverse are reflections over the line \(y=x\). Parallel line functions have inverses that are also lines parallel to \(y=x\).
1Step 1: Find the Inverse of the Function
The given function is \(f(x) = x + 1\). To find the inverse \(f^{-1}(x)\), first replace \(f(x)\) with \(y\), getting \(y = x + 1\). Then, swap \(x\) and \(y\) to get \(x = y + 1\). Solve for \(y\) by subtracting 1 from both sides: \(y = x - 1\). Therefore, the inverse function is \(f^{-1}(x) = x - 1\).
2Step 2: Graph the Function and Its Inverse
To graph \(f(x) = x + 1\) and \(f^{-1}(x) = x - 1\), plot each line on a coordinate plane. The line \(f(x) = x + 1\) has a slope of 1 and a y-intercept at (0, 1). The line \(f^{-1}(x) = x - 1\) has a slope of 1 and a y-intercept at (0, -1). Draw the line \(y = x\), which is a line through the origin with a slope of 1, using dashes or dots. Notice that the lines \(f\) and \(f^{-1}\) are reflections of each other over the line \(y = x\).
3Step 3: Find Inverse of the Function with Parameter b
The function given is \(f(x) = x + b\). To find its inverse, start with \(y = x + b\), swap \(x\) and \(y\) to get \(x = y + b\), and then solve for \(y\) by subtracting \(b\) from both sides: \(y = x - b\). Thus, the inverse function is \(f^{-1}(x) = x - b\).
4Step 4: Relationship Between the Graph of f and Its Inverse
The graph of \(f(x) = x + b\) is a line parallel to the line \(f(x) = x\) and has a y-intercept of \(b\). Its inverse, \(f^{-1}(x) = x - b\), is also a line with the same slope but a y-intercept of \(-b\). Both graphs are reflections of each other over the line \(y = x\).
5Step 5: Conclusion about Inverses of Parallel Line Graphs
If a function has a graph that is a line parallel to the line \(y = x\), its inverse will also be a line parallel to \(f(x) = x\). The inverse graph will cross the \(y\)-axis at a position symmetrical to the graph of \(f(x)\) with respect to the line \(y = x\).
Key Concepts
Graphing Linear FunctionsReflection Over Line y=xLinear Function Inverses
Graphing Linear Functions
Linear functions are the simplest type of functions you can encounter. They are generally of the form \(f(x) = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. This means that the graph of a linear function is always a straight line.
To graph a linear function like \(f(x) = x + 1\), start by identifying the slope and y-intercept:
Linear functions showcase a constant rate of increase or decrease, thus making them quite predictable and easy to sketch.
To graph a linear function like \(f(x) = x + 1\), start by identifying the slope and y-intercept:
- Slope (\(m\)): This indicates the steepness of the line. For \(f(x) = x + 1\), the slope is 1.
- Y-intercept (\(b\)): This is where the line crosses the y-axis. Here, it is 1, meaning the line crosses the y-axis at the point \((0, 1)\).
Linear functions showcase a constant rate of increase or decrease, thus making them quite predictable and easy to sketch.
Reflection Over Line y=x
When graphing a function and its inverse, an essential concept is the reflection over the line \(y = x\). This line acts as a mirror for the function and its inverse.
In the case of the function \(f(x) = x + 1\) and its inverse \(f^{-1}(x) = x - 1\), each point on \(f(x)\) reflected over the line \(y = x\) will land on a corresponding point of \(f^{-1}(x)\).
Here's how the reflection works:
In the case of the function \(f(x) = x + 1\) and its inverse \(f^{-1}(x) = x - 1\), each point on \(f(x)\) reflected over the line \(y = x\) will land on a corresponding point of \(f^{-1}(x)\).
Here's how the reflection works:
- The line \(y = x\) itself is a 45-degree angle line through the origin, with every point on this line satisfying the condition \(y = x\).
- Every point \((a, b)\) on the graph of \(f(x)\) corresponds to the point \((b, a)\) on the graph of its inverse \(f^{-1}(x)\).
Linear Function Inverses
Finding the inverse of a linear function involves swapping roles of the dependent and independent variables. For a function \(f(x) = x + b\), follow these steps to find its inverse:
The significance of a linear function's inverse lies in its graph reflecting across the \(y = x\) line. For functions like \(f(x) = x + b\), both the function and its inverse are parallel to this line, with intercepts symmetrically opposite. The slope remains constant, demonstrating that inverses of linear functions preserve parallelism and symmetry properties.
- Replace \(f(x)\) with \(y\) to get \(y = x + b\).
- Swap \(x\) and \(y\) in the equation, leading to \(x = y + b\).
- Solve for \(y\), resulting in \(y = x - b\).
The significance of a linear function's inverse lies in its graph reflecting across the \(y = x\) line. For functions like \(f(x) = x + b\), both the function and its inverse are parallel to this line, with intercepts symmetrically opposite. The slope remains constant, demonstrating that inverses of linear functions preserve parallelism and symmetry properties.
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