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MAT 201 Quiz 2
February 11, 2005
- Consider the function f (x) = 3 x − 1 x + 2 (a) What is the domain of f?
R\ {− 2 } (b) Evaluate f (2)
f (2) =
- Consider the functions f (x) = 2x + 1 and g(x) = x^2.
(a) Find the composition f ◦ g.
f ◦ g(x) = f (g (x)) = f
x^2
= 2x^2 + 1 (b) Find the composition g ◦ f.
g ◦ f (x) = g (f (x)) = g (2x + 1) = (2x + 1)^2 (c) Find the inverse f −^1 (x). To find the inverse of a function f , substitute y for f (x) and switch the x’s and y’s. So we have x = 2y + 1 When we solve for y we get
y = x − 1 2 and so f −^1 (x) = x − 1 2 (d) Evaluate the difference quotient f (x + h) − f (x) h
f (x + h) − f (x) h
(2(x + h) + 1) − (2x + 1) h = 2 x + 2h + 1 − 2 x − 1 h = 2 h h = 2
- Simplify tan
sin−^1 (x)
If we let y = sin−^1 (x) then we have sin(y) = x. So we are asked to simplify tan(y) where sin(y) = x. If we make a reference triangle, the opposite side will be√ x, and the hypotenuse will be 1. This forces the adjacent side to be 1 − x^2 to satisfy the pythagorean identity. Hence
tan(y) = opposite adjacent
x √ 1 − x^2
- Evaluate
sin−^1
π 6
- Consider the graph of the function f given below:
(a) Find limx→− 2 + f (x). 3 (b) Find limx→− 2 − f (x). 1 (c) Find limx→− 2 f (x). D.N.E. since the above limits are different. (d) Find limx→ 3 − f (x). − 2 (e) Find limx→ 3 + f (x). − 3 (f) Find f (3). − 2
