How can I verify that f(x)=(x+3)/(x-2)and g(x)=(2x+3)/(x-1)are inverse functions (i.e. show the process)?
Also, how can I solve (g o f)^-1, if f(x)=x+4 and g(x)=2x-5?
Thanks!
To verify that two functions, f(x) and g(x), are inverse functions of each other, you need to show that when you compose them and then take the inverse, you get back to the original functions.
1. Verifying f and g as inverse functions:
To check if f(x) and g(x) are inverse functions, you need to show that f(g(x)) = x and g(f(x)) = x for all x in their domains.
a) To find f(g(x)):
Replace x in f(x) with g(x):
f(g(x)) = f((2x+3)/(x-1))
Simplify f(g(x)):
First, find the inverse of g(x):
g⁻¹(x) = (x+5)/2 "by swapping x and y and solving for y"
Substitute g⁻¹(x) into f(g(x)):
f(g(x)) = f(g⁻¹(x)) = f((x+5)/2) = ((x+5)/2 + 3)/((x+5)/2 - 2)
Simplify f(g(x)) further:
f(g(x)) = ((x+5)/2 + 3)/((x+5)/2 - 2) = (x + 5 + 6)/(x + 5 - 4) = (x + 11)/(x + 1)
Now, check if f(g(x)) equals x:
(x + 11)/(x + 1) = x
Cross-multiply:
x + 11 = x(x + 1)
Expand:
x + 11 = x² + x
Rearrange:
x² = 10
This equation is not true for all x, which means f(g(x)) ≠ x. Therefore, f(x) and g(x) are not inverse functions.
b) To find g(f(x)):
Replace x in g(x) with f(x):
g(f(x)) = g(x+4) = 2(x+4) - 5 = 2x + 3
Now, check if g(f(x)) equals x:
2x + 3 = x
Subtract x from both sides:
x + 3 = 0
This equation is also not true for all x, which means g(f(x)) ≠ x. Therefore, f(x) and g(x) are not inverse functions.
2. Solving (g o f)⁻¹:
To find the inverse of the composite function (g o f), follow these steps:
a) Find the composite function (g o f):
(g o f)(x) = g(f(x))
Replace f(x) in g(f(x)):
(g o f)(x) = g(x + 4) = 2(x + 4) - 5 = 2x + 3
b) Find the inverse of (g o f)(x):
To find the inverse, swap x and y and solve for y:
y = 2x + 3
Swap x and y:
x = 2y + 3
Solve for y:
x - 3 = 2y
y = (x - 3)/2
Therefore, the inverse of (g o f) is given by [(g o f)^-1](x) = (x - 3)/2.