Question 1

Find the inverse of the one-to-one function. $y = 7 x$
A) $\quad y = 7 x ^ { 2 }$
B) $\quad y = 7 x$
C) $\quad y = \frac { 7 } { x }$
D) $\quad y = 49 x$
E) $y = \frac { x } { 7 }$

Accepted Answer

To find the inverse of the function $y = 7x$, you swap $x$ and $y$ and solve for $y$. This gives $x = 7y$, which simplifies to $y = \frac{x}{7}$.

Question 2

Determine whether the function is one-to-one. $$y = \frac { 4 } { 7 } x$$
A) Yes, it is one-to-one.
B) No, it isn't one-to-one.

Accepted Answer

Since for any value of $x$, there is only one corresponding value of $y$ (and vice versa), the function passes the horizontal line test, indicating that it is one-to-one.

Question 3

Graph the quadratic function. $f ( x ) = - x ^ { 2 } - 4 x + 5$
A)

B)

C)

D)

Accepted Answer

The answer of Graph the quadratic function. \(f ( x...

Question 4

Let $f ( x ) = \sqrt { x } , g ( x ) = x + 2$. Find the composite function which expresses the given correspondence correctly.
$\sqrt [ 4 ] { x }$
A) $\quad ( f \circ f ) ( x )$
B) $\quad ( g \circ f ) ( x )$
C) $\quad ( g \circ g ) ( x )$
D) $\quad ( f \circ g ) ( x )$
E) none of the above

Accepted Answer

$ ( f \circ f ) ( x ) = f ( f ( x ) ) = f ( \sqrt { x } ) = \sqrt { \sqrt { x } } = \sqrt [ 4 ] { x }$.

Question 5

A taxicab company charges $\$ 5$ for a trip up to 1 mile, and $\$ 4$ for every extra mile (or portion of a mile). Graph the ordered pairs $( m , c )$, where $m$ represents the miles traveled and $c$ represents the cost.
A)

B)

C)

D)

Accepted Answer

For the first mile, the cost is fixed at $\$5$. This means that the graph starts at the point $(1,5)$. For every extra mile, the cost increases by $\$4$. This means that the graph has a slope of $\$4$ for every mile, or $\frac{4}{1}$.

Therefore, we can start at the point $(1,5)$, and use the slope of $\frac{4}{1}$ to find the next point. For example, to find the point that represents a trip of 2 miles, we can add 1 to the $x$-coordinate (since we are traveling an extra mile), and add 4 to the $y$-coordinate (since the cost increases by $\$4$ for every extra mile). This gives us the point $(2,9)$.

Using this method, we can find the following points:

\begin{align*}
(1,5) \
(2,9) \
(3,13) \
(4,17) \
(5,21) \
(6,25) \
\end{align*}

Plotting these on a graph and connecting them with a line gives us choice (A).

Question 6

The function $f ( x ) = \sqrt { x ^ { 2 } - 5 }$ is one-to-one on the domain $x \leq - \sqrt { 5 }$. Find $f ^ { - 1 } ( x )$.
A) $\quad f ^ { - 1 } ( x ) = x ^ { 2 } + 5$
B) $f ^ { - 1 } ( x ) = \sqrt { x ^ { 2 } + 5 }$
C) $f ^ { - 1 } ( x ) = \frac { 1 } { \sqrt { x ^ { 2 } - 5 } }$
D) $\quad f ^ { - 1 } ( x ) = - \sqrt { x ^ { 2 } + 5 }$
E) $f ^ { - 1 } ( x ) = - \sqrt { x ^ { 2 } - 5 }$

Accepted Answer

To find the inverse function $f^{-1}(x)$, we swap $x$ and $y$ in the original function and solve for $y$. Starting with $y = \sqrt{x^2 - 5}$, we swap to get $x = \sqrt{y^2 - 5}$, then square both sides to eliminate the square root, resulting in $x^2 = y^2 - 5$. Solving for $y$, we get $y^2 = x^2 + 5$, and taking the square root gives $y = \pm\sqrt{x^2 + 5}$. Since the domain is $x \leq -\sqrt{5}$, the function is decreasing, and we choose the negative branch to maintain the function's behavior, resulting in $f^{-1}(x) = -\sqrt{x^2 + 5}$.

Question 7

Let $f ( x ) = 2 x - 1 , g ( x ) = 3 x - 2$. Find the domain of the function.
$$( f \cdot g ) ( x )$$
A) $\quad ( - \infty , \infty )$
B) $\quad ( - \infty , 0 ]$
C) $( - \infty , 0 )$
D) $[ 0 , \infty )$
E) $\quad ( 0 , \infty )$

Accepted Answer

The domain of the function is the set of all possible values of $x$ for which $f \cdot g$ is defined.

We have:

$$(f \cdot g)(x) = f(x) \cdot g(x) = (2x-1)(3x-2) = 6x^2-10x-3$$

This is a polynomial function, which is defined for all real values of $x$. Therefore, the domain of the function is $(-\infty,\infty)$.

Thus, the answer is $\boxed{	ext{(A)}}$.

Question 8

Let $f ( x ) = 3 x , g ( x ) = x + 1$. Find the composite function.
$$( g \circ g ) ( x )$$
A) $\quad ( g \circ g ) ( x ) = 3 x + 1$
B) none of the above
C) $\quad ( g \circ g ) ( x ) = 3 x + 3$
D) $\quad ( g \circ g ) ( x ) = 9 x$
E) $\quad ( g \circ g ) ( x ) = x + 2$

Accepted Answer

We have $g(g(x)) = g(x+1) = (x+1)+1 = x+2$. Therefore, the answer is $\boxed{	extbf{(E) } x+2}$.

Question 9

Evaluate the difference quotient for the function. $$f ( x ) = 4 x ^ { 2 } - 5$$
A) $\quad 8 x + 4 h$
B) 1
C) $\quad 8$
D) $4 x + h$
E) 4

Accepted Answer

The difference quotient for a function is given by:

$$\frac{f(x + h) - f(x)}{h}$$

Substitute the given function and simplify:

$$\frac{(4(x+h)^2-5)-(4x^2-5)}{h} = \frac{8xh+4h^2}{h}$$

Simplify further to get:

$$8x + 4h$$

Therefore, the correct choice is A, which matches this expression.

Question 10

The graph of the function $g ( x )$ is a translation of the graph of $f ( x ) = x ^ { 2 }$. Graph the function. $g ( x ) = ( x + 1 ) ^ { 2 } + 3$
A)

B)

C)

D)

Accepted Answer

The answer of The graph of the function \(g (...

Question 11

Let $f ( x ) = \sqrt { x + 1 } , g ( x ) = x ^ { 2 } - 1$. Find the domain of the composite function.
$$( f \circ g ) ( x )$$
A) $\quad ( - \infty , - 1 )$
B) $\quad ( - \infty , - 1 ]$
C) $\quad ( - \infty , \infty )$
D) $\quad ( - 1 , \infty )$
E) $[ - 1 , \infty )$

Accepted Answer

The answer of Let \(f ( x ) = \sqrt...

Question 12

Graph the function. $$y = [ [ 3 x ] ]$$
A)

B)

C)

$\mathrm { d }$.

Accepted Answer

The function is evaluating to the greatest integer less than or equal to $3x$. This means that the output will only change when $3x$ crosses an integer boundary. Therefore, the graph will consist of a series of horizontal line segments, with the lines moving up whenever $3x$ crosses a multiple of 1, and moving down whenever $3x$ crosses a multiple of $-\frac{1}{3}$.
The graph has a slope of $0$ at every integer multiple of $\frac{1}{3}$ (where the function intersects an integer value). 
Using this information, graph the points $(0,0),(0.333,0),(0.666,0),(1,1),(1.333,1), (1.666,1),(2,2),$ etc. and connect them with horizontal line segments.

Note: the correct answer was edited to fix an error in the original coordinates provided for point $(1,1)$.

Question 13

Find the vertex of the parabola. $y = 3 x ^ { 2 } - 2$
A) $\quad ( - 2,0 )$
B) $\quad ( - 2,3 )$
C) $\quad ( 0,3 )$
D) $\quad ( 3 , - 2 )$
E) $\quad ( 0 , - 2 )$

Accepted Answer

The vertex form of a parabola is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex. For $y = 3x^2 - 2$, the equation is already centered at $x = 0$, so $h = 0$, and $k = -2$, making the vertex $(0, -2)$.

Question 14

The amount $A$ of money on deposit for $t$ years in an account earning simple interest is a linear function of $t$. Express that function as an equation if $A = \$ 28$ when $t = 1$ and $A = \$ 30$ when $t = 2$.
A) $A = 2 t - 26$
B) $A = - 2 t + 26$
C) $\quad A = 30 t + 26$
D) $A = 2 t + 26$
E) $A = 30 t - 26$

Accepted Answer

We can use the slope-intercept form of a linear equation, which is
$$A = mt + b,$$
where $m$ is the slope and $b$ is the $y$-intercept. Since the function is linear, we know that the slope is constant. We can find the slope using the given information:
$$m = \frac{	ext{change in }A}{	ext{change in }t} = \frac{\$ 30 - \$ 28}{2 - 1} = \$ 2.$$
Now we can use the point-slope form of a linear equation to find $b$:
$$A - 28 = m(t - 1) \Rightarrow A - 28 = \$ 2(t - 1) \Rightarrow A = 2t - 26.$$
Therefore, the answer is $\boxed{	ext{(D)}}.$

Question 15

A 330-room hotel is two-thirds filled when the nightly room rate is $70.Experience has shown that each $5 increase in cost results in 10 fewer occupied rooms.Find the nightly rate that will maximize
Income. 
A) $\$ 55.00$
B) $\$ 180.00$
C) $\$ 90.00$
D) $\$ 270.00$
E) $ \$ 117.50$

Accepted Answer

To maximize income, we need to find the price point where the product of the number of rooms filled and the price per room is the highest. Let's denote the number of $5 increases from the base price of $70 as $x$. Each increase results in 10 fewer rooms being occupied, so the number of rooms filled can be expressed as $330 - 10x$, and the price per room as $70 + 5x$. The income can then be modeled as $I = (330 - 10x)(70 + 5x)$.Expanding this, we get $I = 23100 + 1650x - 70x - 50x^2$, simplifying to $I = 23100 + 1580x - 50x^2$. To find the maximum income, we take the derivative of $I$ with respect to $x$ and set it to zero. The derivative is $1580 - 100x = 0$, solving for $x$ gives $x = 15.8$.Substituting $x = 15.8$ back into the price equation $70 + 5x$ gives the price that maximizes income: $70 + 5(15.8) = 70 + 79 = \$149$. However, this exact value is not among the options, indicating a need to reevaluate the approach or the interpretation of the options.Given the options, the closest approach to maximizing income based on the relationship between price increases and room occupancy would be to find a balance that doesn't drastically reduce occupancy while increasing the rate. The error in the calculation above suggests revisiting the problem's constraints and the provided options.Upon reevaluation, the mistake lies in the misinterpretation of the options and the calculation error. The correct approach involves understanding that the problem is asking for the optimal price point without requiring exact calculations that lead to an answer not provided in the options. The goal is to understand the relationship between price increases, occupancy, and overall income, which involves finding the point where the decrease in occupancy is offset by the increase in price per room.Given the error in the detailed calculation and the absence of a direct match in the provided options, the correct response should align with the principle of maximizing revenue through strategic pricing, considering the decrease in occupancy with each price increase. The initial setup and exploration aimed to model this relationship, but the final selection should be based on the optimal balance as per the problem's constraints and the options given, which was mistakenly calculated. Revisiting the problem with the correct methodology and considering the constraints and options provided would lead to a different conclusion, focusing on the strategic increase in price that maximizes revenue without significantly reducing occupancy. However, the initial response incorrectly calculated the specific optimal price point, not aligning with the provided options and the correct methodology for solving this type of problem.

Question 16

Find the range of the function $f ( x ) = \frac { 2 x } { x - 8 }$ by finding the domain of $f ^ { - 1 } ( x )$.
A) $\quad ( - \infty , 0 ) \cup ( 0 , \infty )$
B) $\quad ( 0 , \infty )$
C) $( - \infty , \infty )$
D) $\quad ( - \infty , 8 ) \cup ( 8 , \infty )$
E) $\quad ( - \infty , 2 ) \cup ( 2 , \infty )$

Accepted Answer

The answer of Find the range of the function \(f...

Question 17

A pizzeria charges $\$ 12.50$ plus $\$ 0.60$ per topping for a large pizza. Find the cost of a pizza that has 3 toppings.
A) $\quad \$ 10.70$
B) $\$ 14.30$
C) $\$ 15.30$
D) $\$ 15.65$
E) $\$ 10.45$

Accepted Answer

The answer of A pizzeria charges $\$ 12.50$ plus \(\$...

Question 18

Graph the quadratic function. $$f ( x ) = - 3 x ^ { 2 } + 2$$
A)

B)

C)

$\mathrm { d }$.

Accepted Answer

Since the leading coefficient is negative, the graph opens downwards. The vertex of the parabola is at the point $(0,2)$. From there, the curve falls steeply to the left and right. Therefore, option B is the correct graph.

Question 19

Find the inverse of the one-to-one function. $$y = 2 x + 7$$
A) $y = \frac { x + 7 } { 2 }$
B) none of the above
C) $\quad y = \frac { 2 } { x - 7 }$
D) $y = \frac { x - 2 } { 7 }$
E) $y = \frac { x - 7 } { 2 }$

Accepted Answer

To find the inverse, we need to solve for $x$ in terms of $y$. We first isolate the $x$ term: $$y - 7 = 2x.$$ Then, we solve for $x$: $$x = \frac{y-7}{2}.$$ Thus, the inverse of the function is $y=\frac{x-7}{2}$ or $\boxed{	extbf{(E)}}$.

Question 20

At a time t seconds after an object is tossed vertically upward, it reaches a height s in feet given by the equation: $s = 320 t - 16 t ^ { 2 }$
In how many seconds does the object reach its maximum height?
A) 800 sec
B) 1,600 sec
C) 10 sec
D) 5 sec

Accepted Answer

To find the time at which the object reaches its maximum height, we need to find the vertex of the parabolic function given. The vertex has the form (t,s) where t is the time and s is the height reached.

The vertex occurs when the derivative of the function is zero:

s' = 320 - 32t = 0
t = 10

Therefore, the maximum height is reached at t=10 seconds, which is choice D.

Quiz 4: Functions Functions and Function Notation

Find the inverse of the one-to-one function. $y = 7 x$

Determine whether the function is one-to-one. $y = \frac { 4 } { 7 } x$

Graph the quadratic function. $f ( x ) = - x ^ { 2 } - 4 x + 5$

Let $f ( x ) = \sqrt { x } , g ( x ) = x + 2$ . Find the composite function which expresses the given correspondence correctly. $\sqrt [ 4 ] { x }$

A taxicab company charges $\$ 5$ for a trip up to 1 mile, and $\$ 4$ for every extra mile (or portion of a mile) . Graph the ordered pairs $( m , c )$ , where $m$ represents the miles traveled and $c$ represents the cost.

The function $f ( x ) = \sqrt { x ^ { 2 } - 5 }$ is one-to-one on the domain $x \leq - \sqrt { 5 }$ . Find $f ^ { - 1 } ( x )$ .

Let $f ( x ) = 2 x - 1 , g ( x ) = 3 x - 2$ . Find the domain of the function. $( f \cdot g ) ( x )$

Let $f ( x ) = 3 x , g ( x ) = x + 1$ . Find the composite function. $( g \circ g ) ( x )$

Evaluate the difference quotient for the function. $f ( x ) = 4 x ^ { 2 } - 5$

The graph of the function $g ( x )$ is a translation of the graph of $f ( x ) = x ^ { 2 }$ . Graph the function. $g ( x ) = ( x + 1 ) ^ { 2 } + 3$

Let $f ( x ) = \sqrt { x + 1 } , g ( x ) = x ^ { 2 } - 1$ . Find the domain of the composite function. $( f \circ g ) ( x )$

Graph the function. $y = [ [ 3 x ] ]$

Find the vertex of the parabola. $y = 3 x ^ { 2 } - 2$

The amount $A$ of money on deposit for $t$ years in an account earning simple interest is a linear function of $t$ . Express that function as an equation if $A = \$ 28$ when $t = 1$ and $A = \$ 30$ when $t = 2$ .

A 330-room hotel is two-thirds filled when the nightly room rate is $70.Experience has shown that each $5 increase in cost results in 10 fewer occupied rooms.Find the nightly rate that will maximize Income.

Find the range of the function $f ( x ) = \frac { 2 x } { x - 8 }$ by finding the domain of $f ^ { - 1 } ( x )$ .

A pizzeria charges $\$ 12.50$ plus $\$ 0.60$ per topping for a large pizza. Find the cost of a pizza that has 3 toppings.

Graph the quadratic function. $f ( x ) = - 3 x ^ { 2 } + 2$

Find the inverse of the one-to-one function. $y = 2 x + 7$

At a time t seconds after an object is tossed vertically upward, it reaches a height s in feet given by the equation: $s = 320 t - 16 t ^ { 2 }$ In how many seconds does the object reach its maximum height?

Quiz 4: Functions Functions and Function Notation

Find the inverse of the one-to-one function. y=7xy = 7 xy=7x

Determine whether the function is one-to-one. y=47xy = \frac { 4 } { 7 } xy=74​x

Graph the quadratic function. f(x) =−x2−4x+5f ( x ) = - x ^ { 2 } - 4 x + 5f(x) =−x2−4x+5

Let f(x) =x,g(x) =x+2f ( x ) = \sqrt { x } , g ( x ) = x + 2f(x) =x​,g(x) =x+2 . Find the composite function which expresses the given correspondence correctly. x4\sqrt [ 4 ] { x }4x​

A taxicab company charges $5\$ 5$5 for a trip up to 1 mile, and $4\$ 4$4 for every extra mile (or portion of a mile) . Graph the ordered pairs (m,c) ( m , c ) (m,c) , where mmm represents the miles traveled and ccc represents the cost.

The function f(x) =x2−5f ( x ) = \sqrt { x ^ { 2 } - 5 }f(x) =x2−5​ is one-to-one on the domain x≤−5x \leq - \sqrt { 5 }x≤−5​ . Find f−1(x) f ^ { - 1 } ( x ) f−1(x) .

Let f(x) =2x−1,g(x) =3x−2f ( x ) = 2 x - 1 , g ( x ) = 3 x - 2f(x) =2x−1,g(x) =3x−2 . Find the domain of the function. (f⋅g) (x) ( f \cdot g ) ( x ) (f⋅g) (x)

Let f(x) =3x,g(x) =x+1f ( x ) = 3 x , g ( x ) = x + 1f(x) =3x,g(x) =x+1 . Find the composite function. (g∘g) (x) ( g \circ g ) ( x ) (g∘g) (x)

Evaluate the difference quotient for the function. f(x) =4x2−5f ( x ) = 4 x ^ { 2 } - 5f(x) =4x2−5

The graph of the function g(x) g ( x ) g(x) is a translation of the graph of f(x) =x2f ( x ) = x ^ { 2 }f(x) =x2 . Graph the function. g(x) =(x+1) 2+3g ( x ) = ( x + 1 ) ^ { 2 } + 3g(x) =(x+1) 2+3

Let f(x) =x+1,g(x) =x2−1f ( x ) = \sqrt { x + 1 } , g ( x ) = x ^ { 2 } - 1f(x) =x+1​,g(x) =x2−1 . Find the domain of the composite function. (f∘g) (x) ( f \circ g ) ( x ) (f∘g) (x)

Graph the function. y=[[3x]]y = [ [ 3 x ] ]y=[[3x]]

Find the vertex of the parabola. y=3x2−2y = 3 x ^ { 2 } - 2y=3x2−2

The amount AAA of money on deposit for ttt years in an account earning simple interest is a linear function of ttt . Express that function as an equation if A=$28A = \$ 28A=$28 when t=1t = 1t=1 and A=$30A = \$ 30A=$30 when t=2t = 2t=2 .