The point (5,-2) lies on the graph of y=f(x), and assume f^-1(x) exists. Find the corresponding point on eAch graph.
A). Y=f^-1(x)
I get (-5,2)
B). Y=-f^-1(-x)+4
I get (5,2)
C). Y=f^-1(x+2)+3
I get (-7,5)
Did I get these right?
To find the corresponding points on each graph, let's start with the given point (5, -2) that lies on the graph of y = f(x).
A). To find the point on the graph of y = f^(-1)(x), we need to find the inverse of f(x) and substitute x = -2 into it. However, you mentioned assuming f^(-1)(x) exists, so it means it does exist. Therefore, to find the inverse, we swap the variables x and y in the equation y = f(x) and solve for y:
x = f^-1(y)
Now, substitute x = 5:
5 = f^-1(y)
The inverse function evaluated at y = 5 gives us the corresponding x-value:
f^-1(5) = 5
So the corresponding point on the graph of y = f^-1(x) is (-5, 5). Note that there was a small mistake in your answer where you wrote (-5, 2), but it should be (-5, 5).
B). To find the point on the graph of y = -f^(-1)(-x) + 4, we use the same procedure. Start by finding the inverse of f(x) by swapping x and y:
x = -f^(-1)(y) + 4
Now, substitute x = 5:
5 = -f^(-1)(y) + 4
Solving for y, we get:
f^(-1)(y) = -1
So the corresponding y-value is -1. Now, substitute y = -1 into the equation x = -f^(-1)(-x) + 4:
x = -f^(-1)(-(-1)) + 4
x = -f^(-1)(1) + 4
Since we already found the corresponding x-value in part A, which is x = -5, we can substitute that:
x = -(-5) + 4
x = 5 + 4
x = 9
Therefore, the corresponding point on the graph of y = -f^(-1)(-x) + 4 is (9, -1).
C). To find the point on the graph of y = f^(-1)(x+2) + 3, follow the same process. Start by finding the inverse of f(x) by swapping x and y:
x = f^(-1)(y+2) + 3
Now, substitute x = 5:
5 = f^(-1)(y+2) + 3
Solving for y+2, we get:
f^(-1)(y+2) = 2
Now, subtract 2 from both sides:
f^(-1)(y+2) - 2 = 0
So the corresponding y-value is -2. Now, substitute y = -2 into the equation x = f^(-1)(x+2) + 3:
x = f^(-1)(-2+2) + 3
x = f^(-1)(0) + 3
Again, since we found the corresponding x-value in part A, which is x = -5, we can substitute that:
x = -5 + 3
x = -2
Therefore, the corresponding point on the graph of y = f^(-1)(x+2) + 3 is (-2, -2).
In conclusion, your answers were slightly incorrect. The correct corresponding points on each graph are:
A). (5, 5)
B). (9, -1)
C). (-2, -2)