# Properties of Function Worksheet

Properties of Function Worksheet
• Page 1
1.
Choose the graph which represents a function.

 a. Graph -A b. Graph -B c. Graph -C d. Graph -A & Graph -C

#### Solution:

A vertical line intersects the graph -A in only one point , so the given curve is the graph of a function.
[Vertical line test.]

A vertical line intersects the graph -B in two popints , so the given curve is not the graph of a function.
[Vertical line test.]

A vertical line intersects the graph-C in two points, so the given curve is not the graph of a function.
[Vertical line test.]

2.
What is the domain of the function defined by $h$($x$) = $\frac{1}{x}$ + ?
 a. (- ∞, ∞) b. (- ∞, 0) U (4 , ∞) c. (- ∞, 0) U (0, 4 ) U (4 , ∞) d. (- ∞, 0] U [0, 4 ] U [4 , ∞)

#### Solution:

h(x) = 1x + 4x - 4 = 5x - 4x(x - 4)

x ≠ 0 and x ≠ 4.
[The denominator of a fraction must not be zero.]

The domain is (-∞, 0) U (0, 4) U (4, ∞).

3.
Find the domain of the function defined by $g$($x$) = .
 a. (- ∞, - 2 ] U [- 2 , 36] b. (- ∞, - 2 ) U (- 2 , 36] c. [- ∞,- 2 ) U (- 2 , 36] d. (- ∞, - 2 ) U (- 2 , ∞)

#### Solution:

g(x) = 36 - x(x + 2)(x2 + 2)

x + 2 ≠ 0 and x ≠ - 2.
[The denominator of a fraction must not be zero.]

36 - x ≥ 0
[The expression under a radical must not be negative.]

x ≤ 36
[Solve.]

The domain is (-∞, - 2) U (- 2, 36].

4.
A relation $f$, which associates to each element of set A with a unique element of set B is called
 a. A mapping from A$\to$A b. A function from B $\to$ A c. Relation d. Function from A$\to$B

#### Solution:

A relation f , which associates to each element of set A with a unique element of set B is called Function from AB .
[Definition.]

5.
If $f$ is a function from set A to set B, then which of the following is correct?
 a. A is called domain of $f$ and B is called codomain of $f$ b. A is called range of $f$ and B is called domain of $f$ c. A is called codomain of $f$ and B is called domain of $f$ d. A is called domain of $f$ and B is called range of $f$

#### Solution:

If f is a function from set A to set B then A is called domain of f and B is called codomain of f.

6.
Which of the following formulas represents a function?
 a. $y$ = , [4, ∞) b. $y$ = $x$2± 3 c. $y$ = $x$ + 5 d. Both A and C

#### Solution:

For the choice A: y = x - 4 on the interval [4,∞)
Here y is a function of x. Because, the graph of the given formula intersects the vertical line at only one point with x-coordinate 4. Hence, the given formula represents a function.
[Vertical line test.]

For the choice B: y = x2 ± 3
For unique value of x, there exists two values for y (or) the graph of the given equation intersects vertical line at more than one point with x-coordinate 0. Hence, the given formula does not represent a function.
[Vertical line test.]

For the choice C: y = x + 5
The graph of the given equation intersects vertical line at only one point with x- coordinate 0. Hence, the given formula represents a function.
[Vertical line test.]

7.
Choose the formula which is not a function.
 a. $x$ = 2 + $y$ b. $y$ = $x$ + 5 c. $y$ = $x$2 d. $x$ = 2 $y$2

#### Solution:

For the choice A: x = 2 + y y = 2 - x
The graph has unique y value for a unique value of x. Hence, it is a function. In other words, no vertical line intersects the graph in more than one point.
[Vertical line test.]

For the choice B: y = x + 5
The graph has unique y value for a unique value of x. Hence, it is a function. In other words, no vertical line intersects the graph in more than one point.
[Vertical line test.]

For the choice C: y = x2
Here, the graph has two y values for every positive value of x. Hence, it is a function. In other words, no vertical line intersects the graph in more than one point.
[Vertical line test.]

For the choice D: x = 2y2 y = ± x2
Here, the graph has two y values for every positive value of x. Hence, it is not a function. In other words, the vertical line intersects the graph in more than one point.
[Vertical line test.]

8.
Find the domain of the function defined by $f$ ($x$) = .
 a. (- ∞, ∞) b. (- ∞, 3 ] c. [- 3 , ∞] d. [ - 3 , ∞)

#### Solution:

f(x) = x + 3

x + 3 ≥ 0.
[The expression under a radical sign should be positive.]

x ≥ - 3
[Solve.]

Hence, the domain of f is the interval [- 3, ∞).

9.
What is the domain of the function $f$ ($x$) = $\frac{7}{{x}^{2}}$?
 a. [- ∞, 0) $\cup$ (0, ∞) b. ( - ∞, 0) $\cup$ [ 0, ∞) c. (- ∞, 0) $\cup$ (0, ∞) d. (- ∞, 0] $\cup$ (0, ∞)

#### Solution:

f (x) = 7x2

0 is the only number, which is not in the domain of f (x).
[For all values of x except 0, 7x2 is defined.]

Hence, the domain of f (x) is (- ∞, 0) (0, ∞).

10.
Find the range of the function $f$($x$) = 9$x$3
 a. [ - ∞, ∞] b. (- ∞, ∞) c. (- ∞, 0] d. (- ∞, 0) U (0,∞)

#### Solution:

The domain of f(x) = 9x3 is (- ∞, ∞).

For all the real values of x, 9x3 is a real number or 9x3 (- ∞, ∞).

So, the range of f(x) = 9x3 is (- ∞,∞).