How do you graph y= -log (base 5)(-x)?
The written question is: Sketch above equation by first describing the transformations performed on the equation y= log (base 5)(x)
And what is the inverse function? x=log (base 5)(y)?
To graph y = -log₅(-x), we need to first understand the transformations performed on y = log₅(x).
1. Horizontal Reflection: The negative sign in front of log₅(-x) reflects the graph of y = log₅(x) across the y-axis. This means that any points that were originally on the right side of the y-axis will now be on the left side, and vice versa.
2. Vertical Reflection: The negative sign in front of log₅(-x) also reflects the graph vertically. Any points that were originally above the x-axis will now be below it, and any points that were originally below the x-axis will now be above it.
To sketch the graph, we can start by plotting some key points on the original graph of y = log₅(x) and then apply the transformations:
1. Choose some x-values greater than 0 (since log₅(x) is only defined for positive values), calculate the corresponding y-values by evaluating log₅(x), and plot the points.
2. Reflect these points horizontally by changing the sign of their x-coordinates.
3. Reflect the points vertically by changing the sign of their y-coordinates.
Repeat these steps for multiple x-values to get a better understanding of the graph.
Now, let's discuss the inverse function.
To find the inverse function, we need to switch the roles of x and y in the equation y = log₅(x):
x = log₅(y)
Now, solve for y:
5^x = y
The inverse function is given by x = log₅(y), or written in exponential form as y = 5^x.
The inverse function represents the reflection of the original graph across the line y = x. This means that any points on the original graph will have their x and y coordinates swapped on the inverse graph.