A function f is defined by f:x→x+10 over x-8,x not equal to 8.Find
the values of x for which f^-1(x)=x.
remember how I did the last one for you?
original:
y = (x+10)/(x-8)
inverse:
x = (y+10)/(y-8)
xy - 8x = y+10
xy - y = 8x + 10
y(x-1) = (8x+10)/(x-1)
f^-1 (x) = (8x+10)/(x-1)
so we want:
(8x+10)/(x-1) = x
x^2 - x = 8x + 10
x^2 - 9x - 10 = 0
(x-10)(x+1) = 0
x = 10 or x = -1
check:
f^-1 (10) = (80+10)/(10-1) = 90/9 = 10
f^-1 (-1) = (-8+10)/(-1-1) = 2/-2 = -1
To find the values of x for which f^-1(x) equals x, we need to find the inverse of the function f(x) = x + 10.
Step 1: Start by replacing f(x) with y: y = x + 10.
Step 2: Swap x and y in the equation: x = y + 10.
Step 3: Solve the equation for y: y = x - 10.
Step 4: The inverse function is f^-1(x) = x - 10.
Now that we have the inverse function, we can find the values of x for which f^-1(x) equals x by setting them equal to each other:
x - 10 = x.
Step 5: Solve for x:
x - x = 10,
0 = 10.
Since 0 is not equal to 10, there are no values of x that satisfy this equation. Therefore, there are no values of x for which f^-1(x) equals x for the given function f:x→x+10 over x-8, x not equal to 8.