Question 1

Write the domain of the function in interval notation.
-$$k ( x ) = \frac { 1 } { \sqrt { 2 x ^ { 2 } + 3 x - 54 } }$$

Accepted Answer

The domain of $k(x) = \frac{1}{\sqrt{2x^2 + 3x - 54}}$ is determined by the condition that the expression under the square root, $2x^2 + 3x - 54$, must be greater than 0 because the square root of a negative number is not real, and the denominator of a fraction cannot be 0. Factoring the quadratic expression gives $2x^2 + 3x - 54 = (2x - 9)(x + 6)$. Setting each factor greater than 0, we find the intervals where the expression is positive are $x < -6$ and $x > \frac{9}{2}$. Thus, the domain is $(- \infty , - 6 ) \cup \left( \frac { 9 } { 2 } , \infty \right)$.

Question 2

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The time required to drive from Atlanta, Georgia, to Nashville, Tennessee, varies as
the average speed at which a vehicle travels.

Accepted Answer

The answer of Write the word or phrase that best...

Question 3

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-a. Given $y = \frac { 24 } { x }$, evaluate $y$ for the given values of $x$. $x = 1 , x = 2 , x = 3 , x = 4$, and $x = 6 .$
b. How does $y$ change when $x$ is doubled?
c. How does $y$ change when $x$ is tripled?
d. Complete the statement. Given $y = \frac { 24 } { x }$, when $x$ increases, $y$ (increases/decreases) proportionally.
e. Complete the statement. Given $y = \frac { 24 } { x }$, when $x$ decreases, $y$ (increases/decreases) proportionally.

Accepted Answer

The answer of Write the word or phrase that best...

Question 4

Write the domain of the function in interval notation.
-$$p ( x ) = \sqrt { x ^ { 2 } - 14 }$$

Accepted Answer

The domain of the function $p(x) = \sqrt{x^2 - 14}$ consists of all $x$ values for which the expression under the square root, $x^2 - 14$, is non-negative. This occurs when $x^2 \geq 14$, which is true for $x \leq -\sqrt{14}$ and $x \geq \sqrt{14}$. Therefore, the domain in interval notation is $(- \infty, -\sqrt{14}] \cup [\sqrt{14}, \infty)$.

Question 5

Solve the inequality. Write the solution set in interval notation.
-$$\frac { t - 6 } { t ^ { 2 } + 4 } < 0$$

Accepted Answer

The inequality $\frac { t - 6 } { t ^ { 2 } + 4 } < 0$ implies that the numerator and denominator have different signs, since the denominator $t^2 + 4$ is always positive for all real t (as $t^2$ is always non-negative and adding 4 ensures it's strictly positive). Thus, the inequality is negative only when the numerator $t - 6 < 0$, which simplifies to $t < 6$. Therefore, the solution set is all real numbers less than 6, represented in interval notation as $( - \infty , 6 )$.

Question 6

Solve the inequality. Write the solution set in interval notation.
-$$\frac { r + 7 } { r ^ { 2 } + 2 } > 0$$

Accepted Answer

The answer of Solve the inequality. Write the solution set...

Question 7

Solve the inequality. Write the solution set in interval notation.
-$$36 z ^ { 2 } - 9 < 0$$

Accepted Answer

The inequality $36z^2 - 9 < 0$ can be rewritten as $(6z)^2 - 3^2 < 0$, which factors to $(6z - 3)(6z + 3) < 0$, or $(6z - 3)(6z + 3) < 0$. This simplifies to $(z - \frac{1}{2})(z + \frac{1}{2}) < 0$, indicating the solution is between $-\frac{1}{2}$ and $\frac{1}{2}$, hence the interval notation is $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$.

Question 8

Solve the problem.
-A landscaping team plans to build a rectangular garden that is between $240 \mathrm { yd } ^ { 2 }$ and $360 \mathrm { yd } ^ { 2 }$ in area. For aesthetic reasons, they also want the length to be $1.2$ times the width. Determine the restrictions on the width so that the dimensions of the garden will meet the required area. Give exact values and the approximated values to the nearest tenth of a yard.

Accepted Answer

The answer of Solve the problem.
-A landscaping team plans to...

Question 9

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-a. Given y = 2x, evaluate y for the given values of x. $$x = 1 , x = 2 , x = 3 , x = 4 , \text { and } x = 5 \text {. }$$ b. How does y change when x is doubled?
c. How does y change when x is tripled?
d. Complete the statement. Given y = 2x, when x increases, y
(increases/decreases) proportionally.
e. Complete the statement. Given y = 2x, when x decreases, y
(increases/decreases) proportionally.

Accepted Answer

The answer of Write the word or phrase that best...

Question 10

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-If k is a nonzero constant real number, then the statement $$y = \frac { k } { x }$$ implies that y varies as x.

Accepted Answer

The answer of Write the word or phrase that best...

Question 11

Write the domain of the function in interval notation.
-$$h ( x ) = \sqrt { \frac { 3 x } { x + 5 } }$$

Accepted Answer

The answer of Write the domain of the function in...

Question 12

Solve the problem.
-The population $P ( t )$ of a bacteria culture is given by $P ( t ) = - 1,500 t ^ { 2 } + 48,000 t + 18,000$, where $t$ is the time in hours after the culture is started. Determine the time(s) at which the population will be greater than 388,500 organisms.

Accepted Answer

The answer of Solve the problem.
-The population \(P ( t...

Question 13

Solve the inequality. Write the solution set in interval notation.
-$$\frac { 7 } { 1 - x } \leq \frac { 2 } { 3 - x }$$

Accepted Answer

To solve the inequality, cross-multiply to get $7(3-x) \leq 2(1-x)$. Simplifying gives $21 - 7x \leq 2 - 2x$. Rearranging terms, we get $5x \leq 19$, so $x \leq \frac{19}{5}$. The critical points are $x = 1$ and $x = 3$ where the original fractions are undefined. Testing intervals, the solution is $(1, 3) \cup \left[\frac{19}{5}, \infty\right)$.

Question 14

Solve the problem.
-A professional fireworks team shoots an 8-in. mortar straight upwards from ground level with an initial speed of $256 \mathrm { ft } / \mathrm { sec }$.
a. Write a function modeling the vertical position $s ( t )$ (in $\mathrm { ft }$ ) of the shell at a time $t$ seconds after launch.
b. The spectators can see the shell rising once it clears a 768-ft tree line. From what period of time after launch is the shell visible?

Accepted Answer

a. The function $s(t) = -16t^2 + 256t$ models the vertical position of the shell using the formula $s(t) = -16t^2 + vt$, where $v$ is the initial velocity (256 ft/sec) and the acceleration due to gravity is $32 ft/sec^2$ (the negative sign indicates a downward acceleration, hence the $16$ in the formula as $1/2$ of $32$). b. To find when the shell is above 768 ft, set $s(t) = 768$ and solve for $t$, which gives the time interval when the shell is visible above the tree line.

Question 15

Solve the problem.
-The average round trip speed $S$ (in mph) of a vehicle traveling a distance of $d$ miles in each direction is given by $\mathrm { S } = \frac { 2 d } { \frac { d } { r _ { 1 } } + \frac { d } { r _ { 2 } } }$ where $r _ { 1 }$ and $r _ { 2 }$ are the rates of speed for the initial trip and the return trip, respectively. Suppose a motorist travels 100 miles to an event and averages 50 mph for the trip to the event. Determine the average speed necessary for the return trip if the motorist wants the average
Speed for the round trip to be at least 60 mph. Round to one decimal place, if necessary.

Accepted Answer

To achieve an average round trip speed of 60 mph, we use the formula $S = \frac{2d}{\frac{d}{r_1} + \frac{d}{r_2}}$, where $d = 100$ miles, $r_1 = 50$ mph, and $S = 60$ mph. Solving for $r_2$, we find that the motorist must average a speed of at least 75.0 mph on the return trip.

Question 16

Write the domain of the function in interval notation.
-$$h ( a ) = \sqrt { 64 - a ^ { 2 } }$$

Accepted Answer

The domain of the function $h(a) = \sqrt{64 - a^2}$ consists of all values of $a$ for which the expression under the square root, $64 - a^2$, is greater than or equal to 0. This occurs when $a^2 \leq 64$, which simplifies to $-8 \leq a \leq 8$. Therefore, the domain in interval notation is $[ - 8,8 ]$.

Question 17

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The value of k in the variation models y = kx and $$y = \frac { k } { x }$$ is called the of .

Accepted Answer

The answer of Write the word or phrase that best...

Question 18

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-If k is a nonzero constant real number, then the statement y = kx implies that y varies as x.

Accepted Answer

The answer of Write the word or phrase that best...

Question 19

Solve the inequality. Write the solution set in interval notation.
-$4 p ^ { 2 } \geq 17$

Accepted Answer

First, divide both sides of the inequality by 4 to get $p^2 \geq \frac{17}{4}$. Taking the square root of both sides gives $p \geq \frac{\sqrt{17}}{2}$ or $p \leq -\frac{\sqrt{17}}{2}$, which means $p$ is either greater than or equal to $\frac{\sqrt{17}}{2}$ or less than or equal to $-\frac{\sqrt{17}}{2}$. This corresponds to the union of two intervals: $\left( - \infty , - \frac { \sqrt { 17 } } { 2 } \right] \cup \left[ \frac { \sqrt { 17 } } { 2 } , \infty \right)$.

Question 20

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-If y varies directly as two or more other variables such as x and w, then y = kxw, and we say that y
varies as x and w.

Accepted Answer

The answer of Write the word or phrase that best...

Quiz 3: Polynomial and Rational Functions

Write the domain of the function in interval notation. - $k ( x ) = \frac { 1 } { \sqrt { 2 x ^ { 2 } + 3 x - 54 } }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The time required to drive from Atlanta, Georgia, to Nashville, Tennessee, varies as the average speed at which a vehicle travels.

Write the domain of the function in interval notation. - $p ( x ) = \sqrt { x ^ { 2 } - 14 }$

Solve the inequality. Write the solution set in interval notation. - $\frac { t - 6 } { t ^ { 2 } + 4 } < 0$

Solve the inequality. Write the solution set in interval notation. - $\frac { r + 7 } { r ^ { 2 } + 2 } > 0$

Solve the inequality. Write the solution set in interval notation. - $36 z ^ { 2 } - 9 < 0$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If k is a nonzero constant real number, then the statement $y = \frac { k } { x }$ implies that y varies as x.

Write the domain of the function in interval notation. - $h ( x ) = \sqrt { \frac { 3 x } { x + 5 } }$

Solve the problem. -The population $P ( t )$ of a bacteria culture is given by $P ( t ) = - 1,500 t ^ { 2 } + 48,000 t + 18,000$ , where $t$ is the time in hours after the culture is started. Determine the time(s) at which the population will be greater than 388,500 organisms.

Solve the inequality. Write the solution set in interval notation. - $\frac { 7 } { 1 - x } \leq \frac { 2 } { 3 - x }$

Write the domain of the function in interval notation. - $h ( a ) = \sqrt { 64 - a ^ { 2 } }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The value of k in the variation models y = kx and $y = \frac { k } { x }$ is called the of .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If k is a nonzero constant real number, then the statement y = kx implies that y varies as x.

Solve the inequality. Write the solution set in interval notation. - $4 p ^ { 2 } \geq 17$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If y varies directly as two or more other variables such as x and w, then y = kxw, and we say that y varies as x and w.

Quiz 3: Polynomial and Rational Functions

Write the domain of the function in interval notation. - k(x) =12x2+3x−54k ( x ) = \frac { 1 } { \sqrt { 2 x ^ { 2 } + 3 x - 54 } }k(x) =2x2+3x−54​1​

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The time required to drive from Atlanta, Georgia, to Nashville, Tennessee, varies as the average speed at which a vehicle travels.

Write the domain of the function in interval notation. - p(x) =x2−14p ( x ) = \sqrt { x ^ { 2 } - 14 }p(x) =x2−14​

Solve the inequality. Write the solution set in interval notation. - t−6t2+4<0\frac { t - 6 } { t ^ { 2 } + 4 } < 0t2+4t−6​<0

Solve the inequality. Write the solution set in interval notation. - r+7r2+2>0\frac { r + 7 } { r ^ { 2 } + 2 } > 0r2+2r+7​>0

Solve the inequality. Write the solution set in interval notation. - 36z2−9<036 z ^ { 2 } - 9 < 036z2−9<0

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If k is a nonzero constant real number, then the statement y=kxy = \frac { k } { x }y=xk​ implies that y varies as x.

Write the domain of the function in interval notation. - h(x) =3xx+5h ( x ) = \sqrt { \frac { 3 x } { x + 5 } }h(x) =x+53x​​

Solve the inequality. Write the solution set in interval notation. - 71−x≤23−x\frac { 7 } { 1 - x } \leq \frac { 2 } { 3 - x }1−x7​≤3−x2​

Write the domain of the function in interval notation. - h(a) =64−a2h ( a ) = \sqrt { 64 - a ^ { 2 } }h(a) =64−a2​

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The value of k in the variation models y = kx and y=kxy = \frac { k } { x }y=xk​ is called the of .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If k is a nonzero constant real number, then the statement y = kx implies that y varies as x.

Solve the inequality. Write the solution set in interval notation. - 4p2≥174 p ^ { 2 } \geq 174p2≥17

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If y varies directly as two or more other variables such as x and w, then y = kxw, and we say that y varies as x and w.

Write the domain of the function in interval notation. - $k ( x ) = \frac { 1 } { \sqrt { 2 x ^ { 2 } + 3 x - 54 } }$

Write the domain of the function in interval notation. - $p ( x ) = \sqrt { x ^ { 2 } - 14 }$

Solve the inequality. Write the solution set in interval notation. - $\frac { t - 6 } { t ^ { 2 } + 4 } < 0$

Solve the inequality. Write the solution set in interval notation. - $\frac { r + 7 } { r ^ { 2 } + 2 } > 0$

Solve the inequality. Write the solution set in interval notation. - $36 z ^ { 2 } - 9 < 0$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If k is a nonzero constant real number, then the statement $y = \frac { k } { x }$ implies that y varies as x.

Write the domain of the function in interval notation. - $h ( x ) = \sqrt { \frac { 3 x } { x + 5 } }$

Solve the inequality. Write the solution set in interval notation. - $\frac { 7 } { 1 - x } \leq \frac { 2 } { 3 - x }$

Write the domain of the function in interval notation. - $h ( a ) = \sqrt { 64 - a ^ { 2 } }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The value of k in the variation models y = kx and $y = \frac { k } { x }$ is called the of .

Solve the inequality. Write the solution set in interval notation. - $4 p ^ { 2 } \geq 17$