the functions f and g are defined by
f(x)=2/(x+1), x>0 g(x)=1-x^2, xER
a)find value of fg(x)giving your answer in the simplest form.
b) find f^-1(x)
c) solve f(x)= f^-1(x)
a) To find the value of fg(x), we need to first find the expression for fg(x) by substituting g(x) into f(x).
f(x) = 2/(x + 1)
g(x) = 1 - x^2
fg(x) = f(g(x))
Replacing x in f(x) with g(x), we get:
fg(x) = 2/((1 - x^2) + 1)
Simplifying further:
fg(x) = 2/(2 - x^2)
So, the value of fg(x) in the simplest form is 2/(2 - x^2).
b) To find the inverse of f(x), denoted f^(-1)(x), we need to interchange x and y and solve for y.
f(x) = 2/(x + 1)
Interchanging x and y:
x = 2/(y + 1)
Now, let's solve for y:
1/y + 1 = 2/x
1/y = 2/x - 1
1/y = (2 - x)/x
y = x/(2 - x)
So, the inverse of f(x), f^(-1)(x), is given by y = x/(2 - x).
c) To solve f(x) = f^(-1)(x), we need to set f(x) equal to f^(-1)(x) and solve for x.
f(x) = f^(-1)(x)
2/(x + 1) = x/(2 - x)
Now, let's cross-multiply and simplify:
2(2 - x) = x(x + 1)
4 - 2x = x^2 + x
Rearranging the equation:
x^2 + x + 2x + 4 - 2 = 0
x^2 + 3x + 2 = 0
Factoring the quadratic equation:
(x + 1)(x + 2) = 0
Setting each factor equal to zero:
x + 1 = 0 or x + 2 = 0
Solving for x, we get:
x = -1 or x = -2
So, the solutions to f(x) = f^(-1)(x) are x = -1 and x = -2.
To find the value of fg(x) in the simplest form, we need to substitute the expression for g(x) into the function f(x) and simplify.
a) fg(x) means we need to multiply f(x) by g(x):
fg(x) = f(x) * g(x)
Substituting the values of f(x) and g(x):
fg(x) = (2/(x+1)) * (1-x^2)
To simplify this expression, we can multiply the numerators and the denominators:
fg(x) = (2 * (1-x^2)) / (x+1)
Expanding the numerator:
fg(x) = (2 - 2x^2) / (x+1)
This is the simplest form of fg(x).
b) To find the inverse of f(x), we need to swap the roles of x and f(x) and solve for the new f(x).
Let y = f(x):
y = 2/(x+1)
Now, we need to solve this equation for x to find the inverse.
First, let's rearrange the equation:
(x+1)/2 = 1/y
Multiplying both sides by 2:
x+1 = 2/y
Subtracting 1 from both sides:
x = (2/y) - 1
Therefore, the inverse of f(x) is:
f^(-1)(x) = (2/x) - 1
c) To solve f(x) = f^(-1)(x), we need to set the two equations equal to each other and solve for x.
f(x) = f^(-1)(x)
Substituting the expressions:
2/(x+1) = (2/x) - 1
Multiplying both sides by (x+1) and x to eliminate the denominators:
2x = 2(x+1) - x(x+1)
Simplifying:
2x = 2x + 2 - x^2 - x
Rearranging the equation:
0 = x^2 + x - 2
This is a quadratic equation. We can factor it:
0 = (x+2)(x-1)
Setting each factor equal to zero and solving for x:
x+2 = 0 or x-1 = 0
x = -2 or x = 1
Therefore, the solutions to f(x) = f^(-1)(x) are x = -2 and x = 1.