the functions f and gare 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)
fg(x) is the product of the two functions.
for f(x) = 2/(x+1), x>0 and
g(x) = 1-x², x∈ℝ
fg(x) = (2/(x+1))(1-x²)
=2(1+x)(1-x)/(x+1)
Now, what is the domain of fg(x)?
Can you divide both numerator and denominator by (1+x)? Why or why not?
a) To find the value of fg(x), we need to substitute g(x) into f(x) and simplify the expression.
First, let's substitute g(x) into f(x):
fg(x) = f(g(x))
g(x) = 1 - x^2
Substituting g(x) into f(x):
fg(x) = f(1 - x^2)
Now, we substitute f(x) = 2/(x + 1) into fg(x):
fg(x) = 2/(1 - x^2 + 1)
Simplifying the expression:
fg(x) = 2/(2 - x^2)
Therefore, the value of fg(x) in the simplest form is 2/(2 - x^2).
b) To find f^-1(x), we need to swap the roles of x and f(x) and solve for the swapped x.
Let's start with the original equation for f(x):
f(x) = 2/(x + 1)
Swap x and f(x):
x = 2/(f(x) + 1)
Now, solve for f(x):
Multiply both sides by (f(x) + 1):
x(f(x) + 1) = 2
Expand the right side:
xf(x) + x = 2
Rearrange to solve for f(x):
xf(x) = 2 - x
Divide both sides by x:
f(x) = (2 - x)/x
Therefore, f^-1(x) = (2 - x)/x.
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)
2/(x + 1) = (2 - x)/x
First, cross multiply:
2x = (2 - x)(x + 1)
Expand the right side:
2x = 2x + 2 - x^2 - x
Simplify the equation:
0 = -x^2 - x + 2
Rearrange the equation:
x^2 + x - 2 = 0
Now, we can solve this quadratic equation. Factoring or using the quadratic formula will give us the solutions for x.