Sheet 2. Properties of a Function of One Variable

2 Properties of a Function of One Variable

Sheet 2. Properties of a Function of One Variable

Exercise 2.1. Given the function

f (x) = 1 − x 1 + x,

nd: f (0), f (−x), f (x + 1), f (x) + 1, f¡₁

x

¢, _{f (x)}¹ . Exercise 2.2. Given the function

f (x) =

( 2^x for |x| ≤ 2 x²− 1 for |x| > 2 ,

nd: f (−1), f (0), f (2), f (−8), f (8).

Exercise 2.3. Let f (x) = x³−xand g (x) = sin 2x. Find: f ¡ g¡_π

12

¢¢, g (f (1)), g (f (2)), f (f (f (1))).

Exercise 2.4. Find: f (f (x)), g (g (x)), f (g (x)), g (f (x)), where f (x) = x² and g (x) = 2^x. Exercise 2.5. Find the domains of the following functions:

a) f (x) = x²

x + 1 b) f (x) = √⁴

1 − x² c) f (x) = 1

√x²− 4x d) f (x) = (x − 2)

r1 + x 1 − x e) f (x) =√

2 + x − x²+ 1

√x²− 3x f) f(x) = 2^x−13x

g) f (x) = e^x2−x−21 h) f (x) = 1

log (1 − x) +√ x + 2 i) f (x) = log |x| j) f (x) = ln (e^x− e)

k) f (x) = logx2 l) f(x) = arcsin (x + 2) m) f (x) = arcsin 2x

1 + x n) f (x) = arccos 2x

1 + x² o) f (x) = arcsin (1 − x) + log (log x) q) f(x) = arctan 2x

x + 1 Exercise 2.6. Let

A) f (x) = x³ B) f (x) = sin x C) f (x) = 1

x for x 6= 0 Draw the graphs of the following functions:

a) x 7→ f (x) b) x 7→ −f (x) c) x 7→ f (−x) d) x 7→ f (x) − 1 e) x 7→ f (x + 1) f) x 7→ f (2 − x) + 1 g) x 7→ |f (x)| h) x 7→ f (|x|)

Last update: October 1, 2008 1

2 Properties of a Function of One Variable Exercise 2.7. Describe properties of the functions graphed below:

a) b)

c) d)

e) f)

Last update: October 1, 2008 2

2 Properties of a Function of One Variable

Exercise 2.8. Using the graphs describe properties of the given functions:

a) f (x) = |x| b) f (x) = |x| + 1 c) f (x) = |x − 2|

d) f (x) = − |x + 1| e) f (x) = 2 − |x + 1| f) f (x) = |4 − x²| g) f (x) = x²− 3x h) f (x) = (x − 1)²− 4 i) f (x) = 2^x+1 j) f (x) = 2^x− 2 k) f (x) = 3^x−2− 1 l) f (x) = 1 −¡₂

3

¢_x

m) f (x) = log3(x + 2) n) f (x) = log¹₂ (−x) + 1 o) f (x) =

( x + 1 for x < 0 1 − x² for x ≥ 0

Last update: October 1, 2008 3