# HiSET: Math : Domain and range of a function

## Example Questions

### Example Question #1 : Functions And Function Notation

Define functions  and

Give the domain of the composition .

None of the other choices gives the correct response.

Explanation:

The domain of the composition of functions  is that the set of all values  that fall in the domain of such that  falls in the domain of .

, a square root function, whose domain is the set of all values of that make the radicand nonnegative. Since the radicand is itself, the domain of  is simply the set of all nonnegative numbers, or .

, a polynomial function, whose domain is the set of all real numbers. Therefore, the domain is not restricted further by . The domain of is .

### Example Question #1 : Functions And Function Notation

Define functions and .

Give the domain of the composition .

Explanation:

The domain of the composition of functions  is that the set of all values  that fall in the domain of such that  falls in the domain of .

, a rational function, so its domain is the set of all numbers except those that make its denominator zero.  if and only if , making its domain .

, a square root function, so the domain of is the set of all values that make the radicand a nonnegative number. Since the radicand is itself, the domain of is simply the set of all nonnegative numbers, or .

Therefore, we need to further restrict the domain to those values  in that make

Equivalently,

Since 1 is positive, and the quotient is nonnegative as well, it follows that

,

and

.

The domain of is therefore .

### Example Question #2 : Functions And Function Notation

Define functions  and .

Give the domain of the composition .

Explanation:

The domain of the composition of functions  is that the set of all values  that fall in the domain of such that  falls in the domain of .

, a polynomial function, so its domain is the set of all real numbers.

, so the domain of is the set of all values that make the radicand a nonnegative number. Since the radicand is itself, the domain of is simply the set of all nonnegative numbers, or .

Therefore, the domain of is the set of all real numbers such that

.

Equivalently,

To solve this, first, solve the corresponding equation:

Factor the polynomial at left using the difference of squares pattern:

By the Zero Factor Property, either

, in which case ,

or

, in which case .

These are the boundary values of the intervals to be tested. Choose an interior value of each interval and test it in the inequality:

Testing :

- True; include

Testing :

- False; exclude

Testing :

- True; include

Include and . Also include the boundary values, since equality is included in the inequality statement ( ).

The correct domain is .

### Example Question #31 : Algebraic Concepts

Define functions  and .

Give the domain of the composition .

Explanation:

The domain of the composition of functions  is the set of all values  that fall in the domain of  such that  falls in the domain of .

, a square root function, so the domain of  is the set of all values  that make the radicand a nonnegative number. Since the radicand is  itself, the domain of  is simply the set of all nonnegative numbers, or .

, a polynomial function; its domain is the set of all reals, so it does not restrict the domain any further.

The correct domain is .

### Example Question #1 : Domain And Range Of A Function

Define functions and .

Give the domain of the composition .

Explanation:

The domain of the composition of functions is the set of all values  that fall in the domain of such that falls in the domain of .

, a square root function, so the domain of is the set of all values that make the radicand a nonnegative number. Since the radicand is itself, the domain of is simply the set of all nonnegative numbers, or .

, a rational function, so its domain is the set of all numbers except those that make its denominator zero.  if and only if , so from , we exclude only the value so that

;

that is,

Excluding this value from the domain, this leaves as the domain of .

### Example Question #1 : Domain And Range Of A Function

Define functions  and .

Give the domain of the function .

Explanation:

The domain of the sum of two functions is the intersections of the domains of the individual functions. Both and are square root functions, so their radicands must both be positive.

The domain of is the set of all values of such that:

This domain is .

The domain of  is the set of all values of such that:

Subtracting 8 tfrom both sides:

This domain is

The domain of is the intersection of the domains, which is .

### Example Question #1 : Domain And Range Of A Function

Define functions  and

Give the domain of the composition .

None of the other choices gives the correct response.

None of the other choices gives the correct response.

Explanation:

The domain of the composition of functions is the set of all values  that fall in the domain of such that falls in the domain of .

, a polynomial function, so the domain of is equal to the set of all real numbers, . Therefore, places no restrictions on the domain of .

, a square root function, whose domain is the set of all values of that make the radicand nonnegative. Since the radicand is itself, the domain of  is simply the set of all nonnegative numbers, or . Therefore, we must select the set of all in such that , or, equivalently,

However, , so , with equality only if . Therefore, the domain of is the single-element set , which is not among the choices.

### Example Question #1 : Domain And Range Of A Function

Define functions  and .

Give the domain of the function .

Explanation:

The domain of the difference of two functions is the intersections of the domains of the individual functions. Both and are square root functions, so their radicands must both be positive.

The domain of is the set of all values of such that:

Subtracting 8 tfrom both sides:

This domain is

The domain of  is the set of all values of such that:

, or

This domain is .

The domain of  is the intersection of these, .

### Example Question #1 : Domain And Range Of A Function

Define functions  and .

Give the domain of the function .

Explanation:

The domain of the sum of two functions is the intersections of the domains of the individual functions.

Both and are cube root functions. There is no restriction on the value of the radicand of a cube root, so both of these functions have as their domain the set of all real numbers . It follows that also has domain .

### Example Question #1 : Domain And Range Of A Function

Define

Give the range of the function.

Explanation:

The range of a function is the set of all possible values of over its domain.

Since this function is piecewise-defined, it is necessary to examine both parts of the function and extract the range of each.

For , it holds that , so

,

or

.

For , it holds that , so

,

or

The overall range of is the union of these sets, or .