**Unit 23 Exercises**

**1. **Which *f* is not a function from **R**
to **R** in the following equations, where **R** is the set
of real numbers ? Explain why they are not a function.

*f(x) = 1/x**f(x) = y*such that*y*^{2}= x*f(x) = x*^{2}- 1

**2. **Find the domain and range of the following functions.

a) the function that assigns to each bit string (of various lengths) the number of zeros in it.

b) the function that assigns the number of bits left over when a bit string (of various lengths) is split into bytes (which are blocks of 8 bits)

**3. **Determine whether each of the following functions from
**Z** to **Z** is one-to-one, where **Z**
is the set of integers.

a) *f(n) = n + 2*

* *b)* f(n) = n² + n + 1*

* *c)* f(n) = n³ - 1*

**4.** Determine whether each of the following functions from
**Z** to **Z** is onto.

a) *f(n) = n + 2*

* *b)* f(n) = n² + n + 1*

* *c)* f(n) = n³ - 1*

**5.** Determine whether each of the following functions is a bijection
from **R** to **R**.

a) *f(x) = 2x + 3*

b) *f(x) = x² + 2*

**6. **Determine whether each of the following functions
from **R** to **R** is *O(x)*.

*f(x) =*10*f(x) =*3*x*+ 7*f(x) = x*² +*x*+ 1*f(x) =*5*ln x*

**7.** Use the definition of big-oh to show that
*x ^{4}* + 5