1. Let
f(x)=2x+5
g(x)=sqrt(f(x) -2
h(x)= f(g(x))
What is h(2)?
2. Define the function f(x)=2x-5. For what value of x is f(x) equal to f^-1(x)? Thanks!
1. To find h(2), we need to substitute x=2 into the function h(x)=f(g(x)).
First, let's find g(x). We are given that g(x) = sqrt(f(x) - 2). So, substitute f(x) = 2x + 5 into g(x):
g(x) = sqrt((2x + 5) - 2).
Now, substitute x=2:
g(2) = sqrt((2*2 + 5) - 2)
= sqrt(9)
= 3.
Now that we have g(2) = 3, we can substitute it into the function h(x):
h(2) = f(g(2))
= f(3).
Given f(x) = 2x + 5, substitute x=3 into f(x):
h(2) = f(3)
= 2(3) + 5
= 6 + 5
= 11.
Therefore, h(2) = 11.
2. To find the value of x for which f(x) is equal to its inverse f^(-1)(x), we need to solve the equation f(x) = f^(-1)(x).
Given f(x) = 2x - 5, the inverse function can be found by swapping x and f(x) and solving for x:
x = 2f(x) - 5.
Now, substitute f(x) with x and solve for x:
x = 2x - 5
x - 2x = -5
-x = -5
x = 5.
Therefore, the value of x for which f(x) is equal to f^(-1)(x) is x = 5.
g(x) has unbalanced parentheses. I will assume you meant g(x) = √f(x) - 2
h(2) = f(g(2))
= f(√(f(2))-2)
= f(√(9)-2) = f(3-2) = f(1) = 7
f^-1(x) = (x+5)/2, so we need
2x-5 = (x+5)/2
4x-10 = x+5
3x = 15
x = 5
plug in x=5 for both functions to verify that it is correct.