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College Algebra: Real Mathematics, Real People

Ron Larson

Chapter 1

Functions and Their Graphs - all with Video Answers

Educators

KS

Section 1

Graphs of Equations

00:50

Problem 1

Name three types of rigid transformations.

AG
Ankit Gupta
Numerade Educator
01:18

Problem 2

Match the rigid transformation of $y=f(x)$ with the correct representation, where $c>0$
a. $h(x)=f(x)+c$
b. $h(x)=f(x)-c$
c. $$\text { c. } h(x)=f(x-c)$$
d. $h(x)=f(x+c)$
i. horizontal shift $c$ units to the left
ii. vertical shift $c$ units upward
iii. horizontal shift $c$ units to the right
iv. vertical shift $c$ units downward

James Kiss
James Kiss
Numerade Educator
00:44

Problem 3

Fill in the blanks.
A reflection in the $x$ -axis of $y=f(x)$ is represented by $h(x)=$ ________ while a reflection in the $y$ axis of $y=f(x)$ is represented by $h(x)=$ ____.

AG
Ankit Gupta
Numerade Educator
01:14

Problem 4

Fill in the blanks.
A nonrigid transformation of $y=f(x)$ represented by $c f(x)$ is a vertical stretch when ______ and a vertical shrink when ____.

AG
Ankit Gupta
Numerade Educator
02:57

Problem 5

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.
$$\begin{aligned}
&f(x)=x\\
&g(x)=x-4\\
&h(x)=3 x
\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
03:51

Problem 6

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.
$$\begin{aligned}
&f(x)=\frac{1}{2} x\\
&g(x)=\frac{1}{2} x+2\\
&h(x)=4(x-2)
\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
02:44

Problem 7

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.
$$\begin{aligned}
&f(x)=x^{2}\\
&g(x)=x^{2}+2\\
&h(x)=(x-2)^{2}
\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
03:01

Problem 8

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.
$$\begin{aligned}
&f(x)=x^{2}\\
&g(x)=3 x^{2}\\
&h(x)=(x+2)^{2}+1
\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
03:31

Problem 9

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.
$$\begin{aligned}
&f(x)=-x^{2}\\
&g(x)=-x^{2}+1\\
&h(x)=-(x+3)^{2}
\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
04:36

Problem 10

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.
$$\begin{aligned}
&f(x)=(x-2)^{2}\\
&g(x)=(x+2)^{2}+2\\
&h(x)=-(x-2)^{2}-1
\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
02:54

Problem 11

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.
$$\begin{aligned}&f(x)=x^{2}\\&g(x)=\frac{1}{2} x^{2}\\&h(x)=(2 x)^{2}\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
03:59

Problem 12

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.
$$\begin{aligned}
&f(x)=x^{2}\\
&g(x)=\frac{1}{4} x^{2}+2\\
&h(x)=-\frac{1}{4} x^{2}
\end{aligned}$$

Betty Fry
Betty Fry
Numerade Educator
03:57

Problem 13

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.
$$\begin{aligned}
&f(x)=|x|\\
&g(x)=|x|-1\\
&h(x)=3|x-3|
\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
04:16

Problem 14

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.
$$\begin{aligned}
&f(x)=|x|\\
&g(x)=|2 x|\\
&h(x)=-2|x+2|-1
\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
04:23

Problem 15

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.
$$\begin{aligned}
&f(x)=-\sqrt{x}\\
&g(x)=\sqrt{x+1}\\
&h(x)=\sqrt{x-2}+1
\end{aligned}$$

Betty Fry
Betty Fry
Numerade Educator
02:22

Problem 16

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.
$$\begin{aligned}
&f(x)=\sqrt{x}\\
&g(x)=\frac{1}{2} \sqrt{x}\\
&h(x)=-\sqrt{x-4}
\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
02:51

Problem 17

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.
$$\begin{aligned}
&f(x)=\frac{1}{x}\\
&g(x)=\frac{1}{x}-2\\
&h(x)=\frac{1}{x-1}+2
\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
02:45

Problem 18

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.
$$\begin{aligned}
&f(x)=\frac{1}{x}\\
&g(x)=\frac{1}{x}-4\\
&h(x)=\frac{1}{x+3}-1
\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
02:05

Problem 19

Use the graph of $f$ to sketch each graph. To print an enlarged copy of the graph, go to MathGraphs.com.
a. $y=f(x)+2$
b. $y=-f(x)$
$$\text { c. } y=f(x-2)$$
d. $y=f(x+3)$
$$\text { e. } y=2 f(x)$$
$$\text { F. } y=f(-x)$$
$$g \cdot y=f\left(\frac{1}{2} x\right)$$

Jennifer Stoner
Jennifer Stoner
Numerade Educator
04:04

Problem 20

Use the graph of $f$ to sketch each graph. To print an enlarged copy of the graph, go to MathGraphs.com.
a. $y=f(x)-1$
b. $y=f(x+2)$
$$\text { c. } y=f(x-1)$$
d. $y=-f(x-2)$
$$\text { e. } y=f(-x)$$
$$y=\frac{1}{2} f(x)$$
$$\text { g. } y=f(2 x)$$

Betty Fry
Betty Fry
Numerade Educator
01:34

Problem 21

Describe the error in graphing the function.
$$f(x)=(x+1)^{2}$$

AG
Ankit Gupta
Numerade Educator
02:06

Problem 22

Describe the error in graphing the function.
$$f(x)=(x-1)^{2}$$

AG
Ankit Gupta
Numerade Educator
01:09

Problem 23

Compare the graph of the function with the graph of its parent function.
$$y=\sqrt{x}+2$$

AG
Ankit Gupta
Numerade Educator
01:09

Problem 24

Compare the graph of the function with the graph of its parent function.
$$y=\frac{1}{x}-5$$

AG
Ankit Gupta
Numerade Educator
01:05

Problem 25

Compare the graph of the function with the graph of its parent function.
$$y=(x-4)^{3}$$

AG
Ankit Gupta
Numerade Educator
01:04

Problem 26

Compare the graph of the function with the graph of its parent function.
$$y=|x+5|$$

AG
Ankit Gupta
Numerade Educator
01:10

Problem 27

Compare the graph of the function with the graph of its parent function.
$$y=x^{2}-2$$

AG
Ankit Gupta
Numerade Educator
01:07

Problem 28

Compare the graph of the function with the graph of its parent function.
$$y=\sqrt{x-2}$$

AG
Ankit Gupta
Numerade Educator
01:15

Problem 29

Identify the parent function and describe the transformation shown in the graph. Write an equation for the graphed function.

AG
Ankit Gupta
Numerade Educator
01:15

Problem 30

Identify the parent function and describe the transformation shown in the graph. Write an equation for the graphed function.

AG
Ankit Gupta
Numerade Educator
01:15

Problem 31

Identify the parent function and describe the transformation shown in the graph. Write an equation for the graphed function.

AG
Ankit Gupta
Numerade Educator
01:15

Problem 32

Identify the parent function and describe the transformation shown in the graph. Write an equation for the graphed function.

AG
Ankit Gupta
Numerade Educator
01:15

Problem 33

Identify the parent function and describe the transformation shown in the graph. Write an equation for the graphed function.

AG
Ankit Gupta
Numerade Educator
01:15

Problem 34

Identify the parent function and describe the transformation shown in the graph. Write an equation for the graphed function.

AG
Ankit Gupta
Numerade Educator
00:19

Problem 35

Compare the graph of the function with the graph of its parent function.
$$y=-x$$

AG
Ankit Gupta
Numerade Educator
00:27

Problem 36

Compare the graph of the function with the graph of its parent function.
$$y=|-x|$$

AG
Ankit Gupta
Numerade Educator
00:27

Problem 37

Compare the graph of the function with the graph of its parent function.
$$y=(-x)^{2}$$

AG
Ankit Gupta
Numerade Educator
00:24

Problem 38

Compare the graph of the function with the graph of its parent function.
$$y=-x^{3}$$

AG
Ankit Gupta
Numerade Educator
00:18

Problem 39

Compare the graph of the function with the graph of its parent function.
$$y=\frac{1}{-x}$$

AG
Ankit Gupta
Numerade Educator
00:23

Problem 40

Compare the graph of the function with the graph of its parent function.
$$y=-\frac{1}{x}$$

AG
Ankit Gupta
Numerade Educator
00:22

Problem 41

Compare the graph of the function with the graph of its parent function.
$$h(x)=4|x|$$

AG
Ankit Gupta
Numerade Educator
00:36

Problem 42

Compare the graph of the function with the graph of its parent function.
$$p(x)=\frac{1}{2} x^{2}$$

AG
Ankit Gupta
Numerade Educator
00:30

Problem 43

Compare the graph of the function with the graph of its parent function.
$$g(x)=\frac{1}{4} x^{3}$$

AG
Ankit Gupta
Numerade Educator
00:32

Problem 44

Compare the graph of the function with the graph of its parent function.
$$y=2 \sqrt{x}$$

AG
Ankit Gupta
Numerade Educator
00:44

Problem 45

Compare the graph of the function with the graph of its parent function.
$$f(x)=\sqrt{4 x}$$

AG
Ankit Gupta
Numerade Educator
00:32

Problem 46

Compare the graph of the function with the graph of its parent function.
$$y=\left|\frac{1}{2} x\right|$$

AG
Ankit Gupta
Numerade Educator
02:49

Problem 47

Use a graphing utility to graph the three functions in the same viewing window. Describe the graphs of $g$ and $h$ relative to the
graph of $f$. $$\begin{aligned}
&f(x)=x^{3}-3 x^{2}\\
&g(x)=f(x+2)\\
&h(x)=\frac{1}{2} f(x)
\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
01:18

Problem 48

Use a graphing utility to graph the three functions in the same viewing window. Describe the graphs of $g$ and $h$ relative to the graph of $f$.
$$\begin{aligned}
&f(x)=x^{3}-3 x^{2}+2\\
&g(x)=f(x-1)\\
&h(x)=f(3 x)
\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
02:41

Problem 49

$$\begin{aligned}
&f(x)=x^{3}-3 x^{2}\\
&g(x)=-\frac{1}{3} f(x)\\
&h(x)=f(-x)
\end{aligned}$$

Betty Fry
Betty Fry
Numerade Educator
01:15

Problem 50

Use a graphing utility to graph the three functions in the same viewing window. Describe the graphs of $g$ and $h$ relative to the graph of $f$.
$$\begin{aligned}
&f(x)=x^{3}-3 x^{2}+2\\
&g(x)=-f(x)\\
&h(x)=f(2 x)
\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
04:36

Problem 51

a. Identify the parent function $f$
b. Describe the sequence of transformations from $f$ to $g$.
c. Sketch the graph of $g$ by hand.
d. Use function notation to write $g$ in terms of the parent function $f$
$$g(x)=2-(x+5)^{2}$$

Betty Fry
Betty Fry
Numerade Educator
02:52

Problem 52

a. Identify the parent function $f$
b. Describe the sequence of transformations from $f$ to $g$.
c. Sketch the graph of $g$ by hand.
d. Use function notation to write $g$ in terms of the parent function $f$
$$g(x)=(x-10)^{2}+5$$

Betty Fry
Betty Fry
Numerade Educator
03:38

Problem 53

a. Identify the parent function $f$
b. Describe the sequence of transformations from $f$ to $g$.
c. Sketch the graph of $g$ by hand.
d. Use function notation to write $g$ in terms of the parent function $f$
$$g(x)=3+2(x-4)^{2}$$

Betty Fry
Betty Fry
Numerade Educator
03:29

Problem 54

a. Identify the parent function $f$
b. Describe the sequence of transformations from $f$ to $g$.
c. Sketch the graph of $g$ by hand.
d. Use function notation to write $g$ in terms of the parent function $f$
$$g(x)=-\frac{1}{4}(x+2)^{2}-2$$

Betty Fry
Betty Fry
Numerade Educator
03:46

Problem 55

a. Identify the parent function $f$
b. Describe the sequence of transformations from $f$ to $g$.
c. Sketch the graph of $g$ by hand.
d. Use function notation to write $g$ in terms of the parent function $f$
$$g(x)=\frac{1}{3}(x-2)^{3}$$

Betty Fry
Betty Fry
Numerade Educator
03:13

Problem 56

a. Identify the parent function $f$
b. Describe the sequence of transformations from $f$ to $g$.
c. Sketch the graph of $g$ by hand.
d. Use function notation to write $g$ in terms of the parent function $f$
$$g(x)=\frac{1}{2}(x+1)^{3}$$

Betty Fry
Betty Fry
Numerade Educator
03:06

Problem 57

a. Identify the parent function $f$
b. Describe the sequence of transformations from $f$ to $g$.
c. Sketch the graph of $g$ by hand.
d. Use function notation to write $g$ in terms of the parent function $f$
$$g(x)=(x-1)^{3}+2$$

Betty Fry
Betty Fry
Numerade Educator
02:31

Problem 58

a. Identify the parent function $f$
b. Describe the sequence of transformations from $f$ to $g$.
c. Sketch the graph of $g$ by hand.
d. Use function notation to write $g$ in terms of the parent function $f$
$$g(x)=-(x+3)^{3}-10$$

Betty Fry
Betty Fry
Numerade Educator
02:45

Problem 59

a. Identify the parent function $f$
b. Describe the sequence of transformations from $f$ to $g$.
c. Sketch the graph of $g$ by hand.
d. Use function notation to write $g$ in terms of the parent function $f$
$$g(x)=\frac{1}{x+8}-9$$

Betty Fry
Betty Fry
Numerade Educator
03:10

Problem 60

a. Identify the parent function $f$
b. Describe the sequence of transformations from $f$ to $g$.
c. Sketch the graph of $g$ by hand.
d. Use function notation to write $g$ in terms of the parent function $f$
$$g(x)=\frac{1}{x-7}+4$$

Betty Fry
Betty Fry
Numerade Educator
03:50

Problem 61

a. Identify the parent function $f$
b. Describe the sequence of transformations from $f$ to $g$.
c. Sketch the graph of $g$ by hand.
d. Use function notation to write $g$ in terms of the parent function $f$
$$g(x)=-2|x-1|-4$$

Betty Fry
Betty Fry
Numerade Educator
03:16

Problem 62

a. Identify the parent function $f$
b. Describe the sequence of transformations from $f$ to $g$.
c. Sketch the graph of $g$ by hand.
d. Use function notation to write $g$ in terms of the parent function $f$
$$g(x)=\frac{1}{2}|x-2|-3$$

Betty Fry
Betty Fry
Numerade Educator
03:35

Problem 63

a. Identify the parent function $f$
b. Describe the sequence of transformations from $f$ to $g$.
c. Sketch the graph of $g$ by hand.
d. Use function notation to write $g$ in terms of the parent function $f$
$$g(x)=-\frac{1}{2} \sqrt{x+3}-1$$

Betty Fry
Betty Fry
Numerade Educator
03:50

Problem 64

a. Identify the parent function $f$
b. Describe the sequence of transformations from $f$ to $g$.
c. Sketch the graph of $g$ by hand.
d. Use function notation to write $g$ in terms of the parent function $f$
$$g(x)=-3 \sqrt{x+1}-6$$

Betty Fry
Betty Fry
Numerade Educator
00:47

Problem 65

Modeling Data The numbers $N$ (in millions) of households in the United States from 2000 through 2013 are given by the ordered pairs of the form $(t, N(t)),$ where $t=0$ represents $2000 .$ A model for the data is $$N(t)=-0.03(t-26.17)^{2}+126.5$$ (0,104.7)
(1,108.2)
(2,109.3)
(3,111.3)
(4,112.0)
(5,113.3)
(6,114.4)
(7,116.0)
(8,116.8)
(9,117.2)
(10,117.5)
(11,119.9)
(12,121.1)
(13,122.5)
a. Describe the transformation of the parent function $f(t)=t^{2}$ b. Use a graphing utility to graph the model and the data in the same viewing window. c. Rewrite the function so that $t=0$ represents $2009 .$ Explain how you got your answer.

Jennifer Stoner
Jennifer Stoner
Numerade Educator
04:35

Problem 66

The depreciation $D$ (in millions of dollars) of the WD-40 Company assets from 2009 through 2013 can be approximated by the function $$D(t)=1.9 \sqrt{t+3.7}$$ where $t=0$ represents $2009 .$ (Source: WD-40 Company)
a. Describe the transformation of the parent function $f(t)=\sqrt{t}$
b. Use a graphing utility to graph the model over the interval $0 \leq t \leq 4$
c. According to the model, in what year will the depreciation of WD- 40 assets be approximately
6 million dollars?
d. Rewrite the function so that $t=0$ represents 2011 . Explain how you got your answer.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:19

Problem 67

Determine whether the statement is true or false. Justify your answer.The graphs of $f(x)=|x|+6$ and $f(x)=|-x|+6$ are identical.

AG
Ankit Gupta
Numerade Educator
00:24

Problem 67

Determine whether the statement is true or false. Justify your answer. The graph of $y=f(-x)$ is a reflection of the graph of $y=f(x)$ in the $x$ -axis.

AG
Ankit Gupta
Numerade Educator
00:41

Problem 68

Use the fact that the graph of $y=f(x)$ has $x$ -intercepts at $x=2$ and $x=-3$ to find the $x$ -intercepts of the given graph. If not possible, state the reason.
$$y=f(-x)$$

AG
Ankit Gupta
Numerade Educator
00:41

Problem 69

Use the fact that the graph of $y=f(x)$ has $x$ -intercepts at $x=2$ and $x=-3$ to find the $x$ -intercepts of the given graph. If not possible, state the reason.
$$y=f(-x)$$

AG
Ankit Gupta
Numerade Educator
00:21

Problem 70

Use the fact that the graph of $y=f(x)$ has $x$ -intercepts at $x=2$ and $x=-3$ to find the $x$ -intercepts of the given graph. If not possible, state the reason.
$$y=2 f(x)$$

AG
Ankit Gupta
Numerade Educator
00:34

Problem 71

Use the fact that the graph of $y=f(x)$ has $x$ -intercepts at $x=2$ and $x=-3$ to find the $x$ -intercepts of the given graph. If not possible, state the reason.
$$y=f(x)+2$$

AG
Ankit Gupta
Numerade Educator
02:14

Problem 72

Use the fact that the graph of $y=f(x)$ has $x$ -intercepts at $x=2$ and $x=-3$ to find the $x$ -intercepts of the given graph. If not possible, state the reason.
$$y=f(x-3)$$

AG
Ankit Gupta
Numerade Educator
01:21

Problem 73

Determine which equation(s) may be represented by the graph shown. (There may be more than one correct answer.)
a. $f(x)=|x+2|+1$
b. $f(x)=|x-1|+2$
$f(x)=|x-2|+1$
d. $f(x)=2+|x-2|$
e. $f(x)=|(x-2)+1|$
$f(x)=1-|x-2|$

Allison Knapp
Allison Knapp
Numerade Educator
01:21

Problem 74

Determine which equation(s) may be represented by the graph shown. (There may be more than one correct answer.)
a. $f(x)=-\sqrt{x}-4$
$f(x)=-4-\sqrt{x}$
$f(x)=-4-\sqrt{-x}$
d. $f(x)=\sqrt{-x}-4$
e. $f(x)=\sqrt{-x}+4$
$f(x)=\sqrt{x}-4$

Allison Knapp
Allison Knapp
Numerade Educator
01:21

Problem 75

Determine which equation(s) may be represented by the graph shown. (There may be more than one correct answer.)
a. $f(x)=(x-2)^{2}-2$
b. $f(x)=(x+4)^{2}-4$
$f(x)=(x-2)^{2}-4$
d. $f(x)=(x+2)^{2}-4$
$f(x)=4-(x-2)^{2}$
$f(x)=4-(x+2)^{2}$

Allison Knapp
Allison Knapp
Numerade Educator
01:32

Problem 76

$f(x)=-(x-4)^{3}+2$
b. $f(x)=-(x+4)^{3}+2$
$f(x)=-(x-2)^{3}+4$
d. $f(x)=(-x-4)^{3}+2$
$f(x)=(x+4)^{3}+2$
$f(x)=(-x+4)^{3}+2$

Allison Knapp
Allison Knapp
Numerade Educator
01:21

Problem 77

You can use either of two methods to graph a function: plotting points, or translating a parent function as shown in this section. Which method do you prefer to use for each function? Explain.
a. $f(x)=3 x^{2}-4 x+1$
b. $f(x)=2(x-1)^{2}-6$

AG
Ankit Gupta
Numerade Educator
02:32

Problem 78

The graph of $y=f(x)$ passes through the points $(0,1),(1,2),$ and $(2,3) .$ Find the corresponding points on the graph of $y=f(x+2)-1$

AG
Ankit Gupta
Numerade Educator
01:16

Problem 79

Compare the graph of $g(x)=a x^{2}$ with the graph of $f(x)=x^{2}$ whena. $0<a<1$ and b. $a>1$.

Allison Knapp
Allison Knapp
Numerade Educator
03:18

Problem 80

Use the graph of $y=f(x)$ to find the intervals on which each of the Egraphs in (a), (b), and (c) is increasing and decreasing. If not possible, then state the reason.
a. $y=-f(x)$ $$\text
{ b. } y=f(x)-3$$
$$\text { c. } y=f(x-1)$$

KS
Kristan Siegel
Numerade Educator
02:00

Problem 81

Determine whether the lines $L_{1}$ and $L_{2}$ passing through the pairs of points are parallel, perpendicular, or neither.
$$\begin{aligned}
&L_{1}:(-2,-2),(2,10)\\
&L_{2}:(-1,3),(3,9)
\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
02:05

Problem 82

Determine whether the lines $L_{1}$ and $L_{2}$ passing through the pairs of points are parallel, perpendicular, or neither.
$$\begin{aligned}
&L_{1}:(-1,-7),(4,3)\\
&L_{2}:(1,5),(-2,-7)
\end{aligned}$$

AG
Ankit Gupta
Numerade Educator
00:58

Problem 83

Find the domain of the function.
$$f(x)=\frac{4}{9-x}$$

AG
Ankit Gupta
Numerade Educator
01:17

Problem 84

Find the domain of the function.
$$f(x)=\frac{\sqrt{x-5}}{x-7}$$

AG
Ankit Gupta
Numerade Educator
01:30

Problem 85

Find the domain of the function.
$$f(x)=\sqrt{100-x^{2}}$$

AG
Ankit Gupta
Numerade Educator
01:07

Problem 86

Find the domain of the function.
$$f(x)=\sqrt[3]{16-x^{2}}$$

AG
Ankit Gupta
Numerade Educator