Question 1

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all
-domain of $f(x)=(x-3)^{2}$
A) $[3, \infty)$
B) $[0, \infty)$
C) $(3, \infty)$
D) $(-\infty, 0]$
E) $[-3, \infty)$
F) $(-\infty, 3]$
G) $(-\infty, \infty)$
H) $(-\infty, 0)$

Accepted Answer

The function $f(x)=(x-3)^{2}$ is a parabola that opens upwards with its vertex at $x=3$. Since it is a quadratic function, its domain is all real numbers, which is represented by the interval notation $(-\infty, \infty)$.

Question 2

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all
-range of $f(x)=(x-3)^{2}$
A) $[3, \infty)$
B) $[0, \infty)$
C) $(3, \infty)$
D) $(-\infty, 0]$
E) $[-3, \infty)$
F) $(-\infty, 3]$
G) $(-\infty, \infty)$
H) $(-\infty, 0)$

Accepted Answer

The function $f(x)=(x-3)^{2}$ is a parabola opening upwards with its vertex at (3, 0). The lowest value of $f(x)$ is 0, and as $x$ increases or decreases from 3, $f(x)$ increases. Therefore, the range is $[0, \infty)$.

Question 3

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all
-domain of $x=y^{2}+3$
A) $[3, \infty)$
B) $[0, \infty)$
C) $(3, \infty)$
D) $(-\infty, 0]$
E) $[-3, \infty)$
F) $(-\infty, 3]$
G) $(-\infty, \infty)$
H) $(-\infty, 0)$

Accepted Answer

The domain of $x=y^{2}+3$ is all real numbers because $y^2$ is always non-negative, and adding 3 shifts the graph up. The smallest value $x$ can take is 3, when $y=0$, and there's no upper limit, hence $[3, \infty)$.

Question 4

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all
-range of $x=y^{2}+3$
A) $[3, \infty)$
B) $[0, \infty)$
C) $(3, \infty)$
D) $(-\infty, 0]$
E) $[-3, \infty)$
F) $(-\infty, 3]$
G) $(-\infty, \infty)$
H) $(-\infty, 0)$

Accepted Answer

The answer of Match the set described with the correct...

Question 5

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all
-domain of $f(x)=3-\sqrt{x}$
A) $[3, \infty)$
B) $[0, \infty)$
C) $(3, \infty)$
D) $(-\infty, 0]$
E) $[-3, \infty)$
F) $(-\infty, 3]$
G) $(-\infty, \infty)$
H) $(-\infty, 0)$

Accepted Answer

The domain of $f(x) = 3 - \sqrt{x}$ is all $x$ values for which the expression under the square root, $x$, is greater than or equal to 0. This is because you cannot take the square root of a negative number in the real number system. Therefore, the domain is all non-negative numbers, or $[0, \infty)$.

Question 6

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all
-range of $f(x)=\sqrt{3-x}$
A) $[3, \infty)$
B) $[0, \infty)$
C) $(3, \infty)$
D) $(-\infty, 0]$
E) $[-3, \infty)$
F) $(-\infty, 3]$
G) $(-\infty, \infty)$
H) $(-\infty, 0)$

Accepted Answer

The answer of Match the set described with the correct...

Question 7

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all
-domain of $f(x)=\sqrt[3]{x+3}$
A) $[3, \infty)$
B) $[0, \infty)$
C) $(3, \infty)$
D) $(-\infty, 0]$
E) $[-3, \infty)$
F) $(-\infty, 3]$
G) $(-\infty, \infty)$
H) $(-\infty, 0)$

Accepted Answer

The domain of $f(x)=\sqrt[3]{x+3}$ includes all real numbers because the cube root function is defined for all real numbers, including negative numbers, zero, and positive numbers. Therefore, the correct interval notation is $(-\infty, \infty)$, which represents all real numbers.

Question 8

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all
-range of $f(x)=\sqrt[3]{x}-3$
A) $[3, \infty)$
B) $[0, \infty)$
C) $(3, \infty)$
D) $(-\infty, 0]$
E) $[-3, \infty)$
F) $(-\infty, 3]$
G) $(-\infty, \infty)$
H) $(-\infty, 0)$

Accepted Answer

The function $f(x)=\sqrt[3]{x}-3$ involves a cube root, which can take any real number as input, and then shifts the graph down by 3 units. This means the range of the function is all real numbers, represented by $(-\infty, \infty)$.

Question 9

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all
-domain of $f(x)=|x-3|$
A) $[3, \infty)$
B) $[0, \infty)$
C) $(3, \infty)$
D) $(-\infty, 0]$
E) $[-3, \infty)$
F) $(-\infty, 3]$
G) $(-\infty, \infty)$
H) $(-\infty, 0)$

Accepted Answer

The domain of $f(x) = |x-3|$ includes all real numbers, since there are no restrictions on $x$ that would prevent the absolute value function from being defined. Therefore, the correct interval notation is $(-\infty, \infty)$.

Question 10

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all
-range of $f(x)=|x|+3$
A) $[3, \infty)$
B) $[0, \infty)$
C) $(3, \infty)$
D) $(-\infty, 0]$
E) $[-3, \infty)$
F) $(-\infty, 3]$
G) $(-\infty, \infty)$
H) $(-\infty, 0)$

Accepted Answer

The function $f(x) = |x| + 3$ has a minimum value of 3 when $x = 0$, and it increases without bound as $x$ moves away from 0 in either direction. Therefore, its range starts from 3 and extends to infinity, which is represented by the interval notation $[3, \infty)$.

Question 11

The graph of $y=f(x)$ is shown here.
  
Sketch the graph of each of the following. Use ordered pairs to indicate 3 points on the graph.
(a) $y=f(x+3)$
(b) $y=f(x)+3$
(c) $y=f(-x)$
(d) $y=-f(x)$
(e) $y=3 f(x)$
(f) $y=|f(x)|$

Accepted Answer

The answer of The graph of $y=f(x)$ is shown here.
...

Question 12

If the point $(2,7)$ lies on the graph of $y=f(x)$, determine a point on the graph of each equation.
(a) $y=f\left(\frac{1}{2} x\right)$
(b) $y=f(4 x)$

Accepted Answer

The answer of If the point $(2,7)$ lies on the...

Question 13

Graph $y=f(x)$ by hand.
(a) $f(x)=(x-1)^{3}+2$
(b) $f(x)=2 \sqrt{x-3}$

Accepted Answer

The answer of Graph $y=f(x)$ by hand.
(a) $f(x)=(x-1)^{3}+2$
(b) $f(x)=2 \sqrt{x-3}$...

Question 14

Observe the coordinates displayed at the bottom of the given screen showing a portion of the graph $y=f(x)$. Answer each of the following based on your observation.
  
(a) If the graph is symmetric with respect to the $y$-axis, what are the coordinates of another point on the graph?
(b) If the graph is symmetric with respect to the origin, what are the coordinates of another point on the graph?
(c) Suppose the graph is symmetric with respect to the $y$-axis. Sketch a typical viewing window with dimensions $[-5,5]$ by $[0,10]$. Then draw the graph you would expect to see in this window.

Accepted Answer

The answer of Observe the coordinates displayed at the bottom...

Question 15

(a) Write a description that explains how the graph of $y=2 \sqrt{x-1}+3$ can be obtained by translating the graph of $y=\sqrt{x}$.
(b) Sketch by hand the graph of $y=-2|x+2|-3$. State the domain and the range.

Accepted Answer

The answer of (a) Write a description that explains how...

Question 16

Consider the graph of the function shown here.
  
State the interval(s) over which the function is:
(a) increasing
(b) decreasing
(c) constant
(d) continuous
(e) What is the domain of the function?
(f) What is the range of this function?

Accepted Answer

The answer of Consider the graph of the function shown...

Question 17

Solve each of the following analytically, showing all steps. Next graph $y_{1}=|4 x+2|$ and $y_{2}=2$ in the standard viewing window of a graphing calculator. Then state how the graphs support your solution in each case.
(a) $|4 x+2|=2$
(b) $|4 x+2|<2$
(c) $|4 x+2|>2$

Accepted Answer

The answer of Solve each of the following analytically, showing...

Question 18

Given $f(x)=3 x^{2}-2 x-6$ and $g(x)=3 x+5$, find each of the following. Simplify the expression when possible.
(a) $(f-g)(x)$
(b) $\frac{f}{g}(x)$
(c) the domain of $\frac{f}{g}$
(d) $(f \circ g)(x)$
(e) $\frac{f(x+h)-f(x)}{h}(h \neq 0)$

Accepted Answer

The answer of Given $f(x)=3 x^{2}-2 x-6$ and $g(x)=3 x+5$,...

Question 19

Consider the piecewise-defined function defined by $f(x)=\left\{\begin{array}{ll}x^{2}-6 & \text { if } x \leq 1 \ \sqrt{x} & \text { if } x>1\end{array}\right.$.
(a) Graph $f$ by hand.
(b) Use a graphing calculator to obtain an accurate graph in the window $[-5,10]$ by $[-10,10]$.

Accepted Answer

The answer of Consider the piecewise-defined function defined by \(f(x)=\left\{\begin{array}{ll}x^{2}-6...

Question 20

In Fairfield you can go to a coffee shop and pay to use their Internet service. If $x$ represents the number of minutes you are online, where $x>0$, then the function defined by $f(x)=.50 \llbracket x \rrbracket+1.50$ gives the total cost in dollars.
(a) Using dot mode and the window $[0,15]$ by $[0,10]$, graph this function on a graphing calculator.
(b) Use the graph to find the cost of being online for 6.5 minutes.

Accepted Answer

The answer of In Fairfield you can go to a...

Quiz 2: Analysis of Graphs of Functions

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -domain of $f(x) =(x-3) ^{2}$

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -range of $f(x) =(x-3) ^{2}$

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -domain of $x=y^{2}+3$

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -range of $x=y^{2}+3$

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -domain of $f(x) =3-\sqrt{x}$

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -range of $f(x) =\sqrt{3-x}$

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -domain of $f(x) =\sqrt[3]{x+3}$

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -range of $f(x) =\sqrt[3]{x}-3$

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -domain of $f(x) =|x-3|$

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -range of $f(x) =|x|+3$

The graph of $y=f(x)$ is shown here. Sketch the graph of each of the following. Use ordered pairs to indicate 3 points on the graph. (a) $y=f(x+3)$ (b) $y=f(x)+3$ (c) $y=f(-x)$ (d) $y=-f(x)$ (e) $y=3 f(x)$ (f) $y=|f(x)|$

If the point $(2,7)$ lies on the graph of $y=f(x)$ , determine a point on the graph of each equation. (a) $y=f\left(\frac{1}{2} x\right)$ (b) $y=f(4 x)$

Graph $y=f(x)$ by hand. (a) $f(x)=(x-1)^{3}+2$ (b) $f(x)=2 \sqrt{x-3}$

(a) Write a description that explains how the graph of $y=2 \sqrt{x-1}+3$ can be obtained by translating the graph of $y=\sqrt{x}$ . (b) Sketch by hand the graph of $y=-2|x+2|-3$ . State the domain and the range.

Consider the graph of the function shown here. State the interval(s) over which the function is: (a) increasing (b) decreasing (c) constant (d) continuous (e) What is the domain of the function? (f) What is the range of this function?

Solve each of the following analytically, showing all steps. Next graph $y_{1}=|4 x+2|$ and $y_{2}=2$ in the standard viewing window of a graphing calculator. Then state how the graphs support your solution in each case. (a) $|4 x+2|=2$ (b) $|4 x+2|<2$ (c) $|4 x+2|>2$

Given $f(x)=3 x^{2}-2 x-6$ and $g(x)=3 x+5$ , find each of the following. Simplify the expression when possible. (a) $(f-g)(x)$ (b) $\frac{f}{g}(x)$ (c) the domain of $\frac{f}{g}$ (d) $(f \circ g)(x)$ (e) $\frac{f(x+h)-f(x)}{h}(h \neq 0)$

Consider the piecewise-defined function defined by $f(x)=\left\{\begin{array}{ll}x^{2}-6 & \text { if } x \leq 1 \\ \sqrt{x} & \text { if } x>1\end{array}\right.$ . (a) Graph $f$ by hand. (b) Use a graphing calculator to obtain an accurate graph in the window $[-5,10]$ by $[-10,10]$ .

Quiz 2: Analysis of Graphs of Functions

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -domain of f(x) =(x−3) 2f(x) =(x-3) ^{2}f(x) =(x−3) 2

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -range of f(x) =(x−3) 2f(x) =(x-3) ^{2}f(x) =(x−3) 2

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -domain of x=y2+3x=y^{2}+3x=y2+3

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -range of x=y2+3x=y^{2}+3x=y2+3

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -domain of f(x) =3−xf(x) =3-\sqrt{x}f(x) =3−x​

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -range of f(x) =3−xf(x) =\sqrt{3-x}f(x) =3−x​

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -domain of f(x) =x+33f(x) =\sqrt[3]{x+3}f(x) =3x+3​

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -range of f(x) =x3−3f(x) =\sqrt[3]{x}-3f(x) =3x​−3

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -domain of f(x) =∣x−3∣f(x) =|x-3|f(x) =∣x−3∣

Match the set described with the correct interval notation. Choices may be used once, more than once, or not at all -range of f(x) =∣x∣+3f(x) =|x|+3f(x) =∣x∣+3

If the point (2,7)(2,7)(2,7) lies on the graph of y=f(x)y=f(x)y=f(x) , determine a point on the graph of each equation. (a) y=f(12x)y=f\left(\frac{1}{2} x\right)y=f(21​x) (b) y=f(4x)y=f(4 x)y=f(4x)

Graph y=f(x)y=f(x)y=f(x) by hand. (a) f(x)=(x−1)3+2f(x)=(x-1)^{3}+2f(x)=(x−1)3+2 (b) f(x)=2x−3f(x)=2 \sqrt{x-3}f(x)=2x−3​

(a) Write a description that explains how the graph of y=2x−1+3y=2 \sqrt{x-1}+3y=2x−1​+3 can be obtained by translating the graph of y=xy=\sqrt{x}y=x​ . (b) Sketch by hand the graph of y=−2∣x+2∣−3y=-2|x+2|-3y=−2∣x+2∣−3 . State the domain and the range.