Question 1

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 $A = \$ 57$ when $t = 7$ and $A = \$ 27$ when $t = 1$.

Accepted Answer

The function for simple interest is $A = P + Prt$, where $P$ is the principal amount, $r$ is the rate of interest per year, and $t$ is the time in years. Given $A = 57$ when $t = 7$ and $A = 27$ when $t = 1$, we can find the slope ($m$) of the line as $\frac{57 - 27}{7 - 1} = \frac{30}{6} = 5$. This represents the rate of change of $A$ with respect to $t$, which is essentially the interest per year. Using one of the points to solve for the y-intercept ($b$), $27 = 5(1) + b$, we find $b = 22$. Thus, the equation is $A = 5t + 22$.

Question 2

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

Accepted Answer

D

Question 3

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

Accepted Answer

C

Question 4

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

Accepted Answer

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

Question 5

A 270-room hotel is two-thirds filled when the nightly room rate is $100.Experience has shown that each $10 increase in cost results in 20 fewer occupied rooms.Find the nightly rate that will maximize
Income.

Accepted Answer

The answer of A 270-room hotel is two-thirds filled when...

Question 6

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

Accepted Answer

$( f \circ f ) ( x )$ means applying $f$ twice. Since $f(x) = \sqrt{x}$, applying it twice gives $\sqrt{\sqrt{x}} = \sqrt[4]{x}$, which matches the given correspondence.

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

To find the domain of $( f \cdot g ) ( x )$, we need to find the values of $x$ for which the multiplication of $f(x)$ and $g(x)$ is defined.

$( f \cdot g ) ( x )$ is given by:
$$( f \cdot g ) ( x ) = f(x) \cdot g(x) = (2x - 1) \cdot (3x - 2)$$

For the multiplication to be defined, both the factors on the right-hand side must be defined.

The domain of $f(x)$ is all real numbers (since it is a linear function). 
The domain of $g(x)$ is also all real numbers (since it is a linear function).

Therefore, the domain of $( f \cdot g ) ( x )$ is all real numbers, which is represented as $( -\infty, \infty )$.

Hence, the correct answer is D.

Question 10

Evaluate the difference quotient for the function. $$f ( x ) = 3 x ^ { 2 } - 20$$

Accepted Answer

The difference quotient is given by the formula $\frac{f(x+h)-f(x)}{h}$. For $f(x) = 3x^2 - 20$, we calculate $f(x+h) = 3(x+h)^2 - 20 = 3(x^2 + 2xh + h^2) - 20 = 3x^2 + 6xh + 3h^2 - 20$. Subtracting $f(x) = 3x^2 - 20$ from this gives $3x^2 + 6xh + 3h^2 - 20 - (3x^2 - 20) = 6xh + 3h^2$. Dividing by $h$ gives $6x + 3h$, which corresponds to choice E.

Question 11

Graph the rational function. $f ( x ) = \frac { 5 x ^ { 3 } + 4 x } { 2 x }$Note that the numerator and denominator of the fraction share a common factor.

Accepted Answer

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

Question 12

Determine whether the function is one-to-one. $$y = \frac { 2 } { 5 } x$$

Accepted Answer

A function is one-to-one if no two different inputs produce the same output. In other words, every output is paired with only one input. For the given function, every input x is paired with a unique output y, so it is one-to-one. Suppose we have two different inputs $x_1$ and $x_2$ such that $y_1=y_2$. Then, $\frac{2}{5}x_1=\frac{2}{5}x_2$. Multiplying by $\frac{5}{2}$ on each side, we get $x_1=x_2$, which implies that $y_1$ and $y_2$ were produced by different input values. Thus, the function is one-to-one.

Question 13

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

Accepted Answer

The given quadratic function is in the form $f(x) = ax^2 + b$, where $a = -4$ and $b=2$. Since $a < 0$, the graph of the quadratic function is a downwards facing parabola. The vertex of the parabola is given by $(h,k)$, where $h = -\frac{b}{2a}$ and $k = f(h)$. Substituting the values of $a$ and $b$, we get $h = -\frac{2}{-8} = \frac{1}{4}$ and $k = f\left(\frac{1}{4}\right) = -4 \cdot \left(\frac{1}{4}\right)^2 + 2 = \frac{3}{2}$. Therefore, the vertex is at $\left(\frac{1}{4},\frac{3}{2}\right)$. Since the coefficient of $x^2$ is $-4$, the parabola is relatively narrow. Combine this information and we can deduce that the correct representation is B.

Question 14

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

Accepted Answer

$$(g \circ g) (x) = g ( g ( x ) ) = g ( x + 1 ) = ( x + 1 )+1 = x+2$$

Question 15

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

Accepted Answer

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

Question 16

A taxicab company charges $\$ 6$ for a trip up to 1 mile, and $\$ 5$ 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

- For up to 1 mile, the cost is $6$, which means the point $(1,6)$ is on the graph.
- For every extra mile, the cost is $5$, which means each additional mile will contribute a slope of $5$ to the graph. Thus, for any $m>1$, we have $c=5(m-1)+6=5m+1$. This gives us the points $(2,11), (3,16), (4,21),$ and so on.

Therefore, the graph passes through the points $(1,6),(2,11),(3,16),(4,21),$ and so on, which is option A.

Question 17

At a time $t$ seconds after an object is tossed vertically upward, it reaches a height $s$ in feet given by the equation:
$$s = 128 t - 16 t ^ { 2 }$$
In how many seconds does the object reach its maximum height?

Accepted Answer

To find the time at which the object reaches maximum height, we need to find the vertex of the parabola defined by the given equation. We can do this using the vertex formula:
$$t_{\text{max}} = - \frac {b} {2a} = - \frac {128} {2(-16)} = 4$$
Therefore, the object reaches its maximum height after 4 seconds (choice C).

Question 18

Find the vertex of the parabola. $$y = 2 x ^ { 2 } - 5$$

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 $y = 2x^2 - 5$, it is already in a form where $h = 0$ and $k = -5$, making the vertex $(0, -5)$.

Question 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

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

Quiz 4: Functions Functions and Function Notation

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 $A = \$ 57$ when $t = 7$ and $A = \$ 27$ when $t = 1$ .

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

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

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

A 270-room hotel is two-thirds filled when the nightly room rate is $100.Experience has shown that each $10 increase in cost results in 20 fewer occupied rooms.Find the nightly rate that will maximize Income.

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

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

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

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

Evaluate the difference quotient for the function. $f ( x ) = 3 x ^ { 2 } - 20$

Graph the rational function. $f ( x ) = \frac { 5 x ^ { 3 } + 4 x } { 2 x }$ Note that the numerator and denominator of the fraction share a common factor.

Determine whether the function is one-to-one. $y = \frac { 2 } { 5 } x$

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

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

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

A taxicab company charges $\$ 6$ for a trip up to 1 mile, and $\$ 5$ 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.

At a time $t$ seconds after an object is tossed vertically upward, it reaches a height $s$ in feet given by the equation: $s = 128 t - 16 t ^ { 2 }$ In how many seconds does the object reach its maximum height?

Find the vertex of the parabola. $y = 2 x ^ { 2 } - 5$

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

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

Quiz 4: Functions Functions and Function Notation

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=$57A = \$ 57A=$57 when t=7t = 7t=7 and A=$27A = \$ 27A=$27 when t=1t = 1t=1 .

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−3) 3g ( x ) = ( x - 3 ) ^ { 3 }g(x) =(x−3) 3

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​

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

A 270-room hotel is two-thirds filled when the nightly room rate is $100.Experience has shown that each $10 increase in cost results in 20 fewer occupied rooms.Find the nightly rate that will maximize Income.

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​

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−3) 3−2g ( x ) = ( x - 3 ) ^ { 3 } - 2g(x) =(x−3) 3−2

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

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)

Evaluate the difference quotient for the function. f(x) =3x2−20f ( x ) = 3 x ^ { 2 } - 20f(x) =3x2−20

Graph the rational function. f(x) =5x3+4x2xf ( x ) = \frac { 5 x ^ { 3 } + 4 x } { 2 x }f(x) =2x5x3+4x​ Note that the numerator and denominator of the fraction share a common factor.

Determine whether the function is one-to-one. y=25xy = \frac { 2 } { 5 } xy=52​x

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

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)

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

A taxicab company charges $6\$ 6$6 for a trip up to 1 mile, and $5\$ 5$5 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.

At a time ttt seconds after an object is tossed vertically upward, it reaches a height sss in feet given by the equation: s=128t−16t2s = 128 t - 16 t ^ { 2 }s=128t−16t2 In how many seconds does the object reach its maximum height?

Find the vertex of the parabola. y=2x2−5y = 2 x ^ { 2 } - 5y=2x2−5

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

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)