I was given a few problems to do for my pre-calc class, and I solved pretty much everything on my own. All I need is for someone to check the answers to make sure they're correct. Anyone willing?
1. f(x) = x + 2 and g(x) = x^2 + 2x + 3. Find f(g(x)) and g(f(x)). Find all values of x for which f(g(x)) = g(f(x)).
2. Find the domain of f(x) = sqrt (x+5) / x^2 - 25.
- f(g(x)) = x^2 + 2x + 5.
- g(f(x)) = x^2 + 6x + 11.
- f(g(x)) = g(f(x)), when x = -6/4.
3. The vertex of f(x) is (-2,5) and the point (1,-1) is on f(x). Find f(x), domain of f(x) and the range of f(x).
- The domain is (-5,5) U (5, + infinity).
4. f(x) = x^2 + 2x + 3. Find the average rate of change between x=-2 and x=0.
- f(x) = (5/3)(x+2)-5
- The domain of the function is (- infinity, + infinity).
- The range of the function is (- infinity, + infinity).
5. f(x) = ax^2 + bx + 2. f(1)=4 and f(-1)=-2. Find the value of a and b.
- The average rate of change is 0/2.
6. f(x) = 1/x^2. Find the average rate of change between x=x+h and x=x. Find the slope of f(x) at x=-1. Write an equation of a line tangent to f(x) at x=-1.
- a = 3.
- b= -1. (i'm pretty sure these are both wrong.
7. f(x) = 3 - 2x. Find the inverse function of f(x).
- The average rate of change is 2x/x^4.
- The slope at x=-1 is -2.
- The tangent line at x=-1 is (y-1) = (1/2)(x+1).
8. A wire 10 cm long is cut into two pieces, one of the length x and he other of the length 10-x. Each piece is bent into the shape of a square. Find the area of both squares. What value of x will minimize the total area of the two squares?
- f(x^-1) = (3-x)/2
I know these are a lot to do, but I'm only asking for someone to tell me if I'm correct or not. A simple correct/incorrect answer for each problem will do (except for #8, but that's up to you).
- The area of the square made from wire x is x^2.
- The area of the square made from wire 10-x is 100-20x+x^2.
- The value of x should be 5 in order to minimize the total area of the two squares. (for this one, i got the answer x=5, but in a really strange way. can anyone show me the steps for this problem?)