f(x)=2x+1
Sketch in a single diagram the graphs of y=f(x) and y=f^-1(x), making clear relationship between the graphs.
I outlined the steps for this type of question in your previous post
www.jiskha.com/questions/1801140/How-to-sketch-the-graph-of-y-f-1-x
Are you even reading my replies?
To sketch the graphs of y = f(x) = 2x + 1 and y = f^(-1)(x), we need to follow these steps:
Step 1: Sketch the graph of y = f(x) = 2x + 1:
- Find some points that satisfy the equation.
- Choose different x-values and calculate their corresponding y-values.
- Plot these points on a coordinate plane.
- Connect the points smoothly to form a straight line.
For example, when x = -1, y = 2(-1) + 1 = -1, so one point is (-1, -1).
When x = 0, y = 2(0) + 1 = 1, so another point is (0, 1).
When x = 1, y = 2(1) + 1 = 3, so another point is (1, 3).
Using these points, we can plot the line and extend it as needed. It will go through the points (-1, -1), (0, 1), and (1, 3).
Step 2: Find the inverse function f^(-1)(x):
- Replace f(x) with y: y = 2x + 1.
- Swap x and y: x = 2y + 1.
- Solve for y: x - 1 = 2y.
- Divide both sides by 2: (x - 1) / 2 = y.
This gives us the inverse function, y = f^(-1)(x) = (x - 1) / 2.
Step 3: Sketch the graph of y = f^(-1)(x):
- Find points that satisfy the inverse function equation.
- Choose different x-values and calculate their corresponding y-values.
- Plot these points on the same coordinate plane as the graph of y = f(x).
- Connect the points smoothly to form a line.
For example, when x = -1, y = (-1 - 1) / 2 = -1, so one point is (-1, -1).
When x = 0, y = (0 - 1) / 2 = -1/2, so another point is (0, -1/2).
When x = 1, y = (1 - 1) / 2 = 0, so another point is (1, 0).
Using these points, we can plot the line for the inverse function. It will go through the points (-1, -1), (0, -1/2), and (1, 0).
Step 4: Observe the relationship between the graphs:
- The graph of y = f(x) is a straight line with a positive slope.
- The graph of y = f^(-1)(x) is also a straight line but has a greater slope.
- The graphs are reflections of each other about the line y = x.
- The point (a, b) on the graph of y = f(x) corresponds to the point (b, a) on the graph of y = f^(-1)(x).
By following these steps and sketching the graphs, you will see the relationship between y = f(x) = 2x + 1 and y = f^(-1)(x) = (x - 1) / 2.
To sketch the graphs of y = f(x) and y = f^-1(x) and understand the relationship between them, let's go step by step:
Step 1: Find the inverse function, f^-1(x), of the given function f(x) = 2x + 1.
To find the inverse function, we need to switch the roles of x and y and solve for y. Let's do that:
Let y = f(x)
=> y = 2x + 1
Now, let's switch x and y:
x = 2y + 1
Solve for y:
x - 1 = 2y
y = (x - 1) / 2
So, the inverse function is f^-1(x) = (x - 1) / 2.
Step 2: Graph f(x) = 2x + 1
To graph y = f(x), we can start by noting the slope and y-intercept of the function.
The slope (m) is 2, which means that for every 1 unit increase in x, the corresponding y value increases by 2.
The y-intercept is 1, which is the point where the line crosses the y-axis, or when x = 0.
So, we can plot the y-intercept at (0, 1), and from there, we can use the slope to draw the line. To plot additional points, we can choose any x value and find the corresponding y value using the equation y = 2x + 1. Repeat this step a few times to get more points.
Step 3: Graph f^-1(x) = (x - 1) / 2
To graph y = f^-1(x), we start by noting that this is the inverse function. As a result, the graph of f^-1(x) is the reflection of f(x) over the line y = x.
We can use the graph of f(x) to find corresponding points for f^-1(x). For example, the point (3, 7) on the graph of f(x) will correspond to the point (7, 3) on the graph of f^-1(x). Plot these points and repeat the process for a few more points to get a better idea of the shape of f^-1(x).
Step 4: Sketch the graphs on a single diagram
Now that we have the points for both f(x) and f^-1(x), we can sketch their graphs on the same diagram. The graph of f(x) will be a straight line, and the graph of f^-1(x) will also be a straight line, but it will have a steeper slope because it is the reflection of f(x) over the line y = x.
Remember that the graphs should intersect at the point of reflection, which is the point (1, 1). Also, label the x and y axes on the graph and provide a title so it is clear which function is which.
By following these steps, you should be able to sketch the graphs of y = f(x) and y = f^-1(x) on a single diagram and see the relationship between the two functions.