Question 1

Evaluate the difference quotient for the function. $$f ( x ) = 4 x ^ { 2 } - 14$$

Accepted Answer

The difference quotient for a function is given by 
$$ \frac { f ( x + h ) - f ( x ) } { h } $$
Substitute the given function into this expression and simplify:
$$ \frac { f ( x + h ) - f ( x ) } { h } = \frac { 4 ( x + h ) ^ { 2 } - 14 - ( 4 x ^ { 2 } - 14 ) } { h } $$
$$ = \frac { 4 ( x ^ { 2 } + 2 x h + h ^ { 2 } ) - 4 x ^ { 2 } } { h } = \frac { 8 x h + 4 h ^ { 2 } } { h } = 8 x + 4 h $$
Therefore, the correct choice is D.

Question 2

The graph of the function $f ( x )$ is a translation of the graph of $y = ( x - 1 ) ^ { 10 }$. Graph the function. $f ( x ) = ( - x - 1 ) ^ { 10 }$

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

The function $f ( x ) = \sqrt { x ^ { 2 } - 6 }$ is one-to-one on the domain $x \leq - \sqrt { 6 }$. Find $f ^ { - 1 } ( x )$.

Accepted Answer

The answer of The function \(f ( x ) =...

Question 5

Find the vertex of the parabola. $$y = 2 x ^ { 2 } - 8$$

Accepted Answer

The vertex form of a parabola is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex. For the given equation $y = 2x^2 - 8$, it can be rewritten as $y = 2(x - 0)^2 - 8$, indicating the vertex is at $(0, -8)$.

Question 6

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

Accepted Answer

We can find the vertex of the quadratic function using the formula $x=-\frac{b}{2a}$ and then plot the vertex and another point to sketch the parabola.
In this case, $a=-1$, $b=-3$, and $c=2$.
So, $x=-\frac{-3}{2(-1)}=\frac{3}{2}$ is the x-coordinate of the vertex.
To find the y-coordinate, we substitute $x=\frac{3}{2}$ into the function and get $f(\frac{3}{2})=-\frac{1}{4}$.
Therefore, the vertex is at $\left(\frac{3}{2},-\frac{1}{4}\right)$.
Plotting this point and another point, we get the graph:

![Graph of f(x)=-x^2-3x+2](https://i.imgur.com/9Itqi1S.png)

So, the answer is (C).

Question 7

At a time $t$ seconds after an object is tossed vertically upward, it reaches a height $s$ in feet given by the equation:
$$s = 192 t - 16 t ^ { 2 }$$
In how many seconds does the object reach its maximum height?

Accepted Answer

The answer of At a time $t$ seconds after an...

Question 8

Graph the function.
$$f ( x ) = \frac { x ^ { 2 } - 2 x - 3 } { x ^ { 2 } - 3 x + 5 }$$

Accepted Answer

The answer of Graph the function.
\[f ( x ) =...

Question 9

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

Accepted Answer

The graph of a translation of a function is the graph of the function shifted horizontally or vertically. Since we do not have information about the direction of the translation, we choose the graph that most closely resembles the original function. Choice C is the closest to the graph of $f(x) = x^3$, with a similar symmetry and shape. Choices A, B, and D do not have the same basic shape as $f(x)$.

Question 10

The graph of the function $g ( x )$ is a translation of the graph of $f ( x ) = x ^ { 2 }$. Graph the function. $g ( x ) = ( x + 4 ) ^ { 2 } + 3$

Accepted Answer

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

Question 11

Graph the rational function. $$f ( x ) = \frac { 3 x ^ { 3 } + 2 x } { 5 x }$$
Note that the numerator and denominator of the fraction share a common factor.

Accepted Answer

To graph this rational function, we need to first simplify it: 
$$f(x) = \frac{3x^3 + 2x}{5x} = \frac{x (3x^2 + 2)}{5x} = \frac{3x^2 + 2}{5}$$
Now we can see that the degree of the numerator is 2 and the degree of the denominator is 0 (a constant), so the graph of the function will be a parabola. 
To find the x-intercepts, we set the numerator equal to 0: 
$$3x^2 + 2 = 0$$
$$x^2 = -\frac{2}{3}$$
This has no real solutions, so there are no x-intercepts. 
To find the y-intercept, we set x = 0: 
$$f(0) = \frac{3(0)^2 + 2}{5} = \frac{2}{5}$$
So the y-intercept is (0, 2/5). 
To find the vertical asymptote(s), we set the denominator equal to 0: 
$$5x = 0$$
This has one solution at x = 0, so there is a vertical asymptote at x = 0. 
To find the horizontal asymptote, we look at the degrees of the numerator and denominator. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. 
Putting this together, we can sketch the graph of the function as a parabola with no x-intercepts, a y-intercept at (0, 2/5), and a vertical asymptote at x = 0. The graph should look like Option B.

Question 12

Graph the function. $y = [ [ 2 x ] ]$

Accepted Answer

The answer of Graph the function. \(y = [ [...

Question 13

Find the inverse of the one-to-one function. $$y = 5 x$$

Accepted Answer

To find the inverse of the function $y=5x$, we need to switch the roles of $x$ and $y$, and solve for the new $y$. Doing so, we get $x=5y$, which we can rewrite as $y=\frac{x}{5}$. Therefore, the inverse of the function is $y=\frac{x}{5}$, or answer choice $\boxed{	extbf{(D)}}$.

Question 14

Find the inverse of the one-to-one function. $$y = 3 x + 9$$

Accepted Answer

To find the inverse of a one-to-one function, we switch the x and y and solve for y. So we have $$x = 3y+9$$ Solving for y, we get $$y=\frac{x-9}{3}$$which matches option D. Therefore, the answer is D.

Question 15

A pizzeria charges $\$ 6.00$ plus $\$ 0.65$ per topping for a large pizza. Find the cost of a pizza that has 2 toppings.

Accepted Answer

The answer of A pizzeria charges $\$ 6.00$ plus \(\$...

Question 16

A taxicab company charges $\$ 3$ for a trip up to 1 mile, and $\$ 2$ 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.

Accepted Answer

The answer of A taxicab company charges $\$ 3$ for...

Question 17

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 $A = \$ 85$ when $t = 8$ and $A = \$ 40$ when $t = 3$.

Accepted Answer

The answer of The amount $A$ of money on deposit...

Question 18

A 330-room hotel is two-thirds filled when the nightly room rate is $100.Experience has shown that each $15 increase in cost results in 30 fewer occupied rooms.Find the nightly rate that will maximize
Income.

Accepted Answer

The answer of A 330-room hotel is two-thirds filled when...

Question 19

Find the range of the function $f ( x ) = \frac { 4 x } { x - 5 }$ by finding the domain of $f ^ { - 1 } ( x )$.

Accepted Answer

To find the range of $f(x) = \frac{4x}{x-5}$, we find the domain of its inverse $f^{-1}(x)$. To do this, we first express $y = \frac{4x}{x-5}$ and then solve for $x$ in terms of $y$. Rearranging, we get $xy - 5y = 4x$, leading to $x(y - 4) = 5y$, and thus $x = \frac{5y}{y-4}$. The domain of this inverse function, which corresponds to the range of the original function, excludes the value that makes the denominator zero, i.e., $y \neq 4$. Therefore, the range of $f(x)$ is $(-\infty, 4) \cup (4, \infty)$.

Question 20

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

Accepted Answer

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

Quiz 4: Functions Functions and Function Notation

Evaluate the difference quotient for the function. $f ( x ) = 4 x ^ { 2 } - 14$

The graph of the function $f ( x )$ is a translation of the graph of $y = ( x - 1 ) ^ { 10 }$ . Graph the function. $f ( x ) = ( - x - 1 ) ^ { 10 }$

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

The function $f ( x ) = \sqrt { x ^ { 2 } - 6 }$ is one-to-one on the domain $x \leq - \sqrt { 6 }$ . Find $f ^ { - 1 } ( x )$ .

Find the vertex of the parabola. $y = 2 x ^ { 2 } - 8$

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

At a time $t$ seconds after an object is tossed vertically upward, it reaches a height $s$ in feet given by the equation: $s = 192 t - 16 t ^ { 2 }$ In how many seconds does the object reach its maximum height?

Graph the function. $f ( x ) = \frac { x ^ { 2 } - 2 x - 3 } { x ^ { 2 } - 3 x + 5 }$

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

The graph of the function $g ( x )$ is a translation of the graph of $f ( x ) = x ^ { 2 }$ . Graph the function. $g ( x ) = ( x + 4 ) ^ { 2 } + 3$

Graph the rational function. $f ( x ) = \frac { 3 x ^ { 3 } + 2 x } { 5 x }$ Note that the numerator and denominator of the fraction share a common factor.

Graph the function. $y = [ [ 2 x ] ]$

Find the inverse of the one-to-one function. $y = 5 x$

Find the inverse of the one-to-one function. $y = 3 x + 9$

A pizzeria charges $\$ 6.00$ plus $\$ 0.65$ per topping for a large pizza. Find the cost of a pizza that has 2 toppings.

A taxicab company charges $\$ 3$ for a trip up to 1 mile, and $\$ 2$ 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 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 $A = \$ 85$ when $t = 8$ and $A = \$ 40$ when $t = 3$ .

A 330-room hotel is two-thirds filled when the nightly room rate is $100.Experience has shown that each $15 increase in cost results in 30 fewer occupied rooms.Find the nightly rate that will maximize Income.

Find the range of the function $f ( x ) = \frac { 4 x } { x - 5 }$ by finding the domain of $f ^ { - 1 } ( x )$ .

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

Quiz 4: Functions Functions and Function Notation

Evaluate the difference quotient for the function. f(x) =4x2−14f ( x ) = 4 x ^ { 2 } - 14f(x) =4x2−14

The graph of the function f(x) f ( x ) f(x) is a translation of the graph of y=(x−1) 10y = ( x - 1 ) ^ { 10 }y=(x−1) 10 . Graph the function. f(x) =(−x−1) 10f ( x ) = ( - x - 1 ) ^ { 10 }f(x) =(−x−1) 10

Graph the quadratic function. f(x) =−3x2+2f ( x ) = - 3 x ^ { 2 } + 2f(x) =−3x2+2

The function f(x) =x2−6f ( x ) = \sqrt { x ^ { 2 } - 6 }f(x) =x2−6​ is one-to-one on the domain x≤−6x \leq - \sqrt { 6 }x≤−6​ . Find f−1(x) f ^ { - 1 } ( x ) f−1(x) .

Find the vertex of the parabola. y=2x2−8y = 2 x ^ { 2 } - 8y=2x2−8

Graph the quadratic function. f(x) =−x2−3x+2f ( x ) = - x ^ { 2 } - 3 x + 2f(x) =−x2−3x+2

At a time ttt seconds after an object is tossed vertically upward, it reaches a height sss in feet given by the equation: s=192t−16t2s = 192 t - 16 t ^ { 2 }s=192t−16t2 In how many seconds does the object reach its maximum height?

Graph the function. f(x) =x2−2x−3x2−3x+5f ( x ) = \frac { x ^ { 2 } - 2 x - 3 } { x ^ { 2 } - 3 x + 5 }f(x) =x2−3x+5x2−2x−3​

The graph of the function g(x) g ( x ) g(x) is a translation of the graph of f(x) =x3f ( x ) = x ^ { 3 }f(x) =x3 . Graph the function.

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+4) 2+3g ( x ) = ( x + 4 ) ^ { 2 } + 3g(x) =(x+4) 2+3

Graph the rational function. f(x) =3x3+2x5xf ( x ) = \frac { 3 x ^ { 3 } + 2 x } { 5 x }f(x) =5x3x3+2x​ Note that the numerator and denominator of the fraction share a common factor.

Graph the function. y=[[2x]]y = [ [ 2 x ] ]y=[[2x]]

Find the inverse of the one-to-one function. y=5xy = 5 xy=5x

Find the inverse of the one-to-one function. y=3x+9y = 3 x + 9y=3x+9

A pizzeria charges $6.00\$ 6.00$6.00 plus $0.65\$ 0.65$0.65 per topping for a large pizza. Find the cost of a pizza that has 2 toppings.

A taxicab company charges $3\$ 3$3 for a trip up to 1 mile, and $2\$ 2$2 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 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=$85A = \$ 85A=$85 when t=8t = 8t=8 and A=$40A = \$ 40A=$40 when t=3t = 3t=3 .

A 330-room hotel is two-thirds filled when the nightly room rate is $100.Experience has shown that each $15 increase in cost results in 30 fewer occupied rooms.Find the nightly rate that will maximize Income.

Find the range of the function f(x) =4xx−5f ( x ) = \frac { 4 x } { x - 5 }f(x) =x−54x​ by finding the domain of f−1(x) f ^ { - 1 } ( x ) f−1(x) .

The graph of the function g(x) g ( x ) g(x) is a translation of the graph of f(x) =x3f ( x ) = x ^ { 3 }f(x) =x3 . Graph the function. g(x) =(x−1) 3−2g ( x ) = ( x - 1 ) ^ { 3 } - 2g(x) =(x−1) 3−2

Evaluate the difference quotient for the function. $f ( x ) = 4 x ^ { 2 } - 14$

The graph of the function $f ( x )$ is a translation of the graph of $y = ( x - 1 ) ^ { 10 }$ . Graph the function. $f ( x ) = ( - x - 1 ) ^ { 10 }$

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

The function $f ( x ) = \sqrt { x ^ { 2 } - 6 }$ is one-to-one on the domain $x \leq - \sqrt { 6 }$ . Find $f ^ { - 1 } ( x )$ .

Find the vertex of the parabola. $y = 2 x ^ { 2 } - 8$

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

At a time $t$ seconds after an object is tossed vertically upward, it reaches a height $s$ in feet given by the equation: $s = 192 t - 16 t ^ { 2 }$ In how many seconds does the object reach its maximum height?

Graph the function. $f ( x ) = \frac { x ^ { 2 } - 2 x - 3 } { x ^ { 2 } - 3 x + 5 }$

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

The graph of the function $g ( x )$ is a translation of the graph of $f ( x ) = x ^ { 2 }$ . Graph the function. $g ( x ) = ( x + 4 ) ^ { 2 } + 3$

Graph the rational function. $f ( x ) = \frac { 3 x ^ { 3 } + 2 x } { 5 x }$ Note that the numerator and denominator of the fraction share a common factor.

Graph the function. $y = [ [ 2 x ] ]$

Find the inverse of the one-to-one function. $y = 5 x$

Find the inverse of the one-to-one function. $y = 3 x + 9$

A pizzeria charges $\$ 6.00$ plus $\$ 0.65$ per topping for a large pizza. Find the cost of a pizza that has 2 toppings.

A taxicab company charges $\$ 3$ for a trip up to 1 mile, and $\$ 2$ 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 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 $A = \$ 85$ when $t = 8$ and $A = \$ 40$ when $t = 3$ .

Find the range of the function $f ( x ) = \frac { 4 x } { x - 5 }$ by finding the domain of $f ^ { - 1 } ( x )$ .

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