Exer ise 2.2

Sheet 2. Properties of a Fun tion of One Variable

Exer ise 2.1. Given the fun tion

f(x) = 1 − x 1 + x,

nd: f(0)^,f(−x)^,f(x + 1)^,f(x) + 1^,f _x¹

,

1 f(x)^.

Exer ise 2.2. Given the fun tion

f(x) =

( 2^{x for} |x| ≤ 2 x² − 1 ^for |x| > 2 ,

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

Exer ise2.3. Letf(x) = x³−x^andg(x) = sin 2x^{. Find:} f g ₁₂^π

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

Exer ise 2.4. Find: f(f (x))^,g(g (x))^,f(g (x))^,g(f (x))^{, where} f(x) = x^{2 and} g(x) = 2^x.

Exer ise 2.5. Find the domainsof the following fun tions:

a)f(x) = x²

x+ 1 ^b)f(x) = √⁴

1 − x²

)f(x) = 1

√x²− 4x

d)f(x) = (x − 2)r 1 + 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) = log_x2 ^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

Exer ise 2.6. Let

A) f(x) = x^{3 B)} f(x) = sin x ^C) f(x) = 1

x ^forx6= 0

Drawthe graphsof the following fun tions:

a)x7→ f (x) ^b)x7→ −f (x) ⁾x7→ f (−x) ^d)x7→ f (x) − 1

e)x7→ f (x + 1) ^f) x7→ f (2 − x) + 1 ^g) x7→ |f (x)| ^h)x7→ f (|x|)

Exer ise 2.7. Des ribe properties of the fun tions graphed below:

a) b)

) d)

e) f)

Exer ise 2.8. Usingthe graphsdes ribeproperties of the given fun tions:

a)f(x) = |x| ^b)f(x) = |x| + 1 ⁾ 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 − ²₃
x

m)f(x) = log₃(x + 2) ⁿ⁾f(x) = log¹

2 (−x) + 1 ^o) f(x) =

( x+ 1 ^for x <0 1 − x^{2 for} x≥ 0