True or False: The graph of a function and its reciprocal are reflections of each other in the x-axis.
Please explain the answer.
Of course not!
Consider y=x and y = 1/x
Not even the same shape.
False. The graph of a function and its reciprocal are not reflections of each other in the x-axis.
To understand this, let's first define what a reciprocal is. The reciprocal of a number is obtained by dividing 1 by that number. In the context of functions, to find the reciprocal of a function, you would divide each output value by 1.
Now, let's consider an example to illustrate this. Suppose we have the function f(x) = 2x. To find the reciprocal of this function, we divide each output value (or y-value) by 1.
Reciprocal of f(x) = 1/f(x) = 1/(2x)
If we graph these two functions on the same coordinate system, we would see that they are not reflections of each other in the x-axis. In fact, their graphs will usually look quite different.
For example, let's plot the original function f(x) = 2x and its reciprocal 1/(2x) on a graph:
Original function f(x) = 2x:
- When x = 0, f(x) = 0
- When x = 1, f(x) = 2
- When x = -1, f(x) = -2
Reciprocal function 1/(2x):
- When x = 0, 1/(2x) is undefined (division by zero is not allowed)
- When x = 1, 1/(2x) = 1/2
- When x = -1, 1/(2x) = -1/2
As you can see, the graphs of the original function and its reciprocal do not mirror each other across the x-axis. Therefore, the statement "The graph of a function and its reciprocal are reflections of each other in the x-axis" is false.