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Magazine Chapter 9: Problem 23
Find the inverse of each function and graph the function and its inverse on the same set of axes. \(f(x)=x+4\)
Short Answer
Expert verified
The inverse of \( f(x) = x + 4 \) is \( f^{-1}(x) = x - 4 \).
Step by step solution
01
Understand the Problem
The given function is \( f(x) = x + 4 \). We need to find its inverse \( f^{-1}(x) \) and graph both the function and its inverse.
02
Replace f(x) with y
For ease of finding the inverse, replace \( f(x) \) with \( y \) resulting in the equation \( y = x + 4 \).
03
Swap x and y
To find the inverse function, swap the variables \( x \) and \( y \) in the equation to get \( x = y + 4 \).
04
Solve for y
Rearrange the equation \( x = y + 4 \) to solve for \( y \) by subtracting 4 from both sides. This gives \( y = x - 4 \).
05
Write the inverse function
Now that we solved for \( y \), the inverse function is \( f^{-1}(x) = x - 4 \).
06
Graph the Functions
Plot the original function \( f(x) = x + 4 \) and the inverse function \( f^{-1}(x) = x - 4 \) on the same coordinate axes. The original function is a line with a slope of 1 and a y-intercept at (0,4), while the inverse is also a line with a slope of 1 but a y-intercept at (0,-4).
07
Verify by Line of Symmetry
Check that the graphs of \( f(x) \) and \( f^{-1}(x) \) are symmetric along the line \( y = x \). This line serves as the axis of symmetry for a function and its inverse.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Graphing
Graphing a function involves plotting points that satisfy the equation on a set of axes. For the function \( f(x) = x + 4 \), we start by identifying key features such as the slope and y-intercept.
This function is a linear equation, meaning it graphs as a straight line. Its slope is 1 (rise over run), meaning for every unit increase in \( x \), \( y \) increases by the same amount. The y-intercept of this function is at (0,4), which is the point where the line crosses the y-axis.
To graph \( f(x) \), we can follow these steps:
This function is a linear equation, meaning it graphs as a straight line. Its slope is 1 (rise over run), meaning for every unit increase in \( x \), \( y \) increases by the same amount. The y-intercept of this function is at (0,4), which is the point where the line crosses the y-axis.
To graph \( f(x) \), we can follow these steps:
- Start at the y-intercept (0,4).
- Use the slope to find another point. From (0,4), move up 1 and right 1 to reach (1,5).
- Draw a line through these points extending to fill the axes.
- Start at (0,-4), the y-intercept.
- Move up 1 and right 1 to another point, such as (1,-3).
- Draw the line through these points.
Symmetry in Graphs
Symmetry is a valuable visual cue in graphing. It allows us to verify the correctness of our inverse function. In particular, a function and its inverse exhibit symmetry about the line \( y = x \).
This means that if we fold the graph along this line, the function \( f(x) \) will overlay perfectly on top of its inverse \( f^{-1}(x) \).
To visualize this:
This means that if we fold the graph along this line, the function \( f(x) \) will overlay perfectly on top of its inverse \( f^{-1}(x) \).
To visualize this:
- Draw the line \( y = x \) on your graph, which is a diagonal line through the origin with slope 1.
- Check that every point on \( f(x) \) has its corresponding point on \( f^{-1}(x) \) mirrored across this line.
Linear Equations
Linear equations are equations of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. These equations graph as straight lines, as seen with our function \( f(x) = x + 4 \) and its inverse.
Linear equations are straightforward, but mastering them builds a strong foundation for understanding more complex functions.
Key properties of linear equations to know:
Linear equations are straightforward, but mastering them builds a strong foundation for understanding more complex functions.
Key properties of linear equations to know:
- Slope (\( m \)): Determines the angle of the line. A positive slope inclines upward to the right, while a negative slope goes downward.
- Y-intercept (\( b \)): The point where the line crosses the y-axis.
- Uniform change: The relationship between \( x \) and \( y \) is constant, so a change in \( x \) results in a consistent change in \( y \).
