Question 1

Graph the greatest integer function.
-$f ( x ) = \llbracket \frac { 1 } { 3 } x \rrbracket$

Accepted Answer

The answer of Graph the greatest integer function.
-\(f ( x...

Question 2

Graph the piecewise function.
-$f ( x ) = \left\{ \begin{array} { l l } 
- 2 x & \text { if } x \leq - 2 \
- 2 x + 2 & \text { if } - 2 < x \leq 2 \
x + 1 & \text { if } x > 2
\end{array} \right.$

Accepted Answer

The answer of Graph the piecewise function.
-\(f ( x )...

Question 3

Find the real zeros of the function, state their multiplicities if it is a number other than 1.
-$f ( x ) = x ^ { 2 } - 25$

Accepted Answer

The function $f(x) = x^2 - 25$ can be factored as $(x - 5)(x + 5)$, giving real zeros at $x = 5$ and $x = -5$, each with multiplicity 1.

Question 4

Find the real zeros of the function, state their multiplicities if it is a number other than 1.
-$f ( x ) = x ^ { 2 } - 16 x + 64$

Accepted Answer

The function $f(x) = x^2 - 16x + 64$ can be factored as $(x - 8)^2$, indicating that x = 8 is a real zero with multiplicity 2.

Question 5

Determine whether the function is an even function, an odd function, or neither.
-$f ( x ) = x ^ { 5 } - x ^ { 4 }$

Accepted Answer

A function is even if $f(-x) = f(x)$ for all $x$ in the domain, and odd if $f(-x) = -f(x)$. For $f(x) = x^5 - x^4$, $f(-x) = (-x)^5 - (-x)^4 = -x^5 - x^4$, which is not equal to $f(x)$ (so it's not even) and not equal to $-f(x)$ (so it's not odd). Therefore, the function is neither even nor odd.

Question 6

Find the real zeros of the function, state their multiplicities if it is a number other than 1.
-$f ( x ) = x ^ { 3 } + x ^ { 2 } - 20 x$

Accepted Answer

First, factor out the greatest common factor, which is $x$, to get $x(x^2 + x - 20)$. Then, factor the quadratic to get $x(x + 5)(x - 4)$. This gives the zeros $x = 0$, $x = -5$, and $x = 4$, each with a multiplicity of 1.

Question 7

Determine whether the function is an even function, an odd function, or neither.
-$f ( x ) = x ^ { 3 } + x ^ { 2 } - 4$

Accepted Answer

A function is even if $f(-x) = f(x)$ for all $x$ in the domain, and odd if $f(-x) = -f(x)$. For $f(x) = x^3 + x^2 - 4$, $f(-x) = (-x)^3 + (-x)^2 - 4 = -x^3 + x^2 - 4$, which is neither $f(x)$ nor $-f(x)$. Therefore, the function is neither even nor odd.

Question 8

Determine whether the function is an even function, an odd function, or neither.
-$f ( x ) = x ^ { 3 } - 4 x$

Accepted Answer

A function is odd if $f(-x) = -f(x)$ for all $x$ in the domain of $f$. For $f(x) = x^3 - 4x$, $f(-x) = (-x)^3 - 4(-x) = -x^3 + 4x = -(x^3 - 4x) = -f(x)$, so $f(x)$ is odd.

Question 9

Determine whether the function is an even function, an odd function, or neither.
-$g ( x ) = e ^ { 3 x } + 5$

Accepted Answer

A function is even if $f(-x) = f(x)$ for all $x$ in the domain, and odd if $f(-x) = -f(x)$. For $g(x) = e^{3x} + 5$, $g(-x) = e^{-3x} + 5 \neq e^{3x} + 5$ and $g(-x) \neq -(e^{3x} + 5)$, so it is neither even nor odd.

Question 10

Graph the greatest integer function.
-$f ( x ) = \llbracket \frac { x + 4 } { 3 } \rrbracket$

Accepted Answer

The answer of Graph the greatest integer function.
-\(f ( x...

Question 11

Solve the problem.
-Sam Wright wants to stock up on cans of cat food which are selling for $0.95 per can. Write a greatest integer function that represents the number of cans he can purchase with x dollars.

Accepted Answer

The greatest integer function that represents the number of cans Sam can purchase with x dollars is $f(x) = \left\lfloor \frac{x}{0.95} \right\rfloor$, because you divide the total amount of money he has by the cost per can to find the maximum number of cans he can buy, and use the floor function to ensure it's an integer.

Question 12

Find the real zeros of the function, state their multiplicities if it is a number other than 1.
-$f ( x ) = 5 ( x + 3 ) ( x + 7 ) ^ { 4 }$

Accepted Answer

The real zeros of the function are found by setting each factor equal to zero. For $5(x + 3)(x + 7)^4$, setting $x + 3 = 0$ gives $x = -3$, and setting $x + 7 = 0$ gives $x = -7$. The multiplicity of the zero is determined by the exponent of its factor, so $x = -7$ has a multiplicity of 4.

Question 13

Find the real zeros of the function, state their multiplicities if it is a number other than 1.
-$f ( x ) = 2 x ^ { 2 } - 26$

Accepted Answer

To find the real zeros of the function $f(x) = 2x^2 - 26$, we set it equal to zero and solve for $x$: $2x^2 - 26 = 0$. Simplifying, we get $x^2 = 13$, which gives us $x = \pm\sqrt{13}$, indicating two distinct real zeros, each with multiplicity 1.

Question 14

Graph the greatest integer function.
-$f ( x ) = \llbracket x - 2 \rrbracket$

Accepted Answer

The answer of Graph the greatest integer function.
-\(f ( x...

Question 15

Find the real zeros of the function, state their multiplicities if it is a number other than 1.
-$f ( x ) = x ^ { 2 } + 10 x + 21$

Accepted Answer

The quadratic equation $x^2 + 10x + 21$ can be factored into $(x + 7)(x + 3) = 0$, giving the real zeros $x = -7$ and $x = -3$, each with multiplicity 1.

Question 16

Solve the problem.
-A paddle boat rental company rents paddle boats by the hour. They charge $7.00 per hour for the first 3 rental hours and $5.00 per hour for each additional hour. Write a piecewise function to represent the cost of renting a
Paddle boat as a function of hours it is rented.

Accepted Answer

The correct piecewise function must account for the initial charge for the first 3 hours ($7 per hour) and then add the additional charge for hours beyond 3 at $5 per hour. For the first 3 hours, the cost is $7 times the number of hours. For hours beyond 3, the total cost is the initial $21 (3 hours at $7 each) plus $5 for each hour beyond 3. This is correctly represented in option C.

Question 17

Determine whether the function is an even function, an odd function, or neither.
-$f ( x ) = 2 x ^ { 2 } + x ^ { 4 }$

Accepted Answer

The function $f(x) = 2x^2 + x^4$ is even because it satisfies the condition for even functions, $f(x) = f(-x)$. Both terms, $2x^2$ and $x^4$, are even powers of $x$, which means their values do not change when $x$ is replaced with $-x$.

Question 18

Determine whether the function is an even function, an odd function, or neither.
-$f ( x ) = - 3 x ^ { 5 } + x ^ { 3 }$

Accepted Answer

A function is odd if $f(-x) = -f(x)$ for all $x$ in the domain of $f$. For $f(x) = -3x^5 + x^3$, $f(-x) = -3(-x)^5 + (-x)^3 = 3x^5 - x^3 = -( -3x^5 + x^3) = -f(x)$, which satisfies the condition for an odd function.

Question 19

Graph the piecewise function.
-$g ( x ) = \left\{ \begin{array} { l l } 
5 x + 5 & \text { if } x \leq 1 \
- \frac { 1 } { 5 } x + \frac { 6 } { 5 } & \text { if } x > 1
\end{array} \right.$

Accepted Answer

The answer of Graph the piecewise function.
-\(g ( x )...

Question 20

Determine whether the function is an even function, an odd function, or neither.
-$g ( x ) = e ^ { x ^ { 3 } } - 4$

Accepted Answer

The function $g(x) = e^{x^3} - 4$ is neither even nor odd. An even function satisfies $g(-x) = g(x)$, and an odd function satisfies $g(-x) = -g(x)$. For this function, $g(-x) = e^{(-x)^3} - 4 = e^{-x^3} - 4$, which is neither equal to $g(x)$ nor $-g(x)$, since $e^{x^3} \neq e^{-x^3}$ for all $x$.

Quiz 10: Characteristics of Functions and Their Graphs

Graph the greatest integer function. - $f ( x ) = \llbracket \frac { 1 } { 3 } x \rrbracket$ $Graph the greatest integer function. - f ( x ) = \llbracket \frac { 1 } { 3 } x \rrbracket A) B) C) D)$

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - $f ( x ) = x ^ { 2 } - 25$

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - $f ( x ) = x ^ { 2 } - 16 x + 64$

Determine whether the function is an even function, an odd function, or neither. - $f ( x ) = x ^ { 5 } - x ^ { 4 }$

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - $f ( x ) = x ^ { 3 } + x ^ { 2 } - 20 x$

Determine whether the function is an even function, an odd function, or neither. - $f ( x ) = x ^ { 3 } + x ^ { 2 } - 4$

Determine whether the function is an even function, an odd function, or neither. - $f ( x ) = x ^ { 3 } - 4 x$

Determine whether the function is an even function, an odd function, or neither. - $g ( x ) = e ^ { 3 x } + 5$

Graph the greatest integer function. - $f ( x ) = \llbracket \frac { x + 4 } { 3 } \rrbracket$ $Graph the greatest integer function. - f ( x ) = \llbracket \frac { x + 4 } { 3 } \rrbracket A) B) C) D)$

Solve the problem. -Sam Wright wants to stock up on cans of cat food which are selling for $0.95 per can. Write a greatest integer function that represents the number of cans he can purchase with x dollars.

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - $f ( x ) = 5 ( x + 3 ) ( x + 7 ) ^ { 4 }$

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - $f ( x ) = 2 x ^ { 2 } - 26$

Graph the greatest integer function. - $f ( x ) = \llbracket x - 2 \rrbracket$ $Graph the greatest integer function. - f ( x ) = \llbracket x - 2 \rrbracket A) B) C) D)$

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - $f ( x ) = x ^ { 2 } + 10 x + 21$

Solve the problem. -A paddle boat rental company rents paddle boats by the hour. They charge $7.00 per hour for the first 3 rental hours and $5.00 per hour for each additional hour. Write a piecewise function to represent the cost of renting a Paddle boat as a function of hours it is rented.

Determine whether the function is an even function, an odd function, or neither. - $f ( x ) = 2 x ^ { 2 } + x ^ { 4 }$

Determine whether the function is an even function, an odd function, or neither. - $f ( x ) = - 3 x ^ { 5 } + x ^ { 3 }$

Determine whether the function is an even function, an odd function, or neither. - $g ( x ) = e ^ { x ^ { 3 } } - 4$

Quiz 10: Characteristics of Functions and Their Graphs

Graph the greatest integer function. - f(x) =⟦13x⟧f ( x ) = \llbracket \frac { 1 } { 3 } x \rrbracketf(x) =[[31​x]]

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - f(x) =x2−25f ( x ) = x ^ { 2 } - 25f(x) =x2−25

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - f(x) =x2−16x+64f ( x ) = x ^ { 2 } - 16 x + 64f(x) =x2−16x+64

Determine whether the function is an even function, an odd function, or neither. - f(x) =x5−x4f ( x ) = x ^ { 5 } - x ^ { 4 }f(x) =x5−x4

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - f(x) =x3+x2−20xf ( x ) = x ^ { 3 } + x ^ { 2 } - 20 xf(x) =x3+x2−20x

Determine whether the function is an even function, an odd function, or neither. - f(x) =x3+x2−4f ( x ) = x ^ { 3 } + x ^ { 2 } - 4f(x) =x3+x2−4

Determine whether the function is an even function, an odd function, or neither. - f(x) =x3−4xf ( x ) = x ^ { 3 } - 4 xf(x) =x3−4x

Determine whether the function is an even function, an odd function, or neither. - g(x) =e3x+5g ( x ) = e ^ { 3 x } + 5g(x) =e3x+5

Graph the greatest integer function. - f(x) =⟦x+43⟧f ( x ) = \llbracket \frac { x + 4 } { 3 } \rrbracketf(x) =[[3x+4​]]

Solve the problem. -Sam Wright wants to stock up on cans of cat food which are selling for $0.95 per can. Write a greatest integer function that represents the number of cans he can purchase with x dollars.

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - f(x) =5(x+3) (x+7) 4f ( x ) = 5 ( x + 3 ) ( x + 7 ) ^ { 4 }f(x) =5(x+3) (x+7) 4

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - f(x) =2x2−26f ( x ) = 2 x ^ { 2 } - 26f(x) =2x2−26

Graph the greatest integer function. - f(x) =⟦x−2⟧f ( x ) = \llbracket x - 2 \rrbracketf(x) =[[x−2]]

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - f(x) =x2+10x+21f ( x ) = x ^ { 2 } + 10 x + 21f(x) =x2+10x+21

Solve the problem. -A paddle boat rental company rents paddle boats by the hour. They charge $7.00 per hour for the first 3 rental hours and $5.00 per hour for each additional hour. Write a piecewise function to represent the cost of renting a Paddle boat as a function of hours it is rented.

Determine whether the function is an even function, an odd function, or neither. - f(x) =2x2+x4f ( x ) = 2 x ^ { 2 } + x ^ { 4 }f(x) =2x2+x4

Determine whether the function is an even function, an odd function, or neither. - f(x) =−3x5+x3f ( x ) = - 3 x ^ { 5 } + x ^ { 3 }f(x) =−3x5+x3

Graph the piecewise function. - g(x) ={5x+5 if x≤1−15x+65 if x>1g ( x ) = \left\{ \begin{array} { l l } 5 x + 5 & \text { if } x \leq 1 \\- \frac { 1 } { 5 } x + \frac { 6 } { 5 } & \text { if } x > 1\end{array} \right.g(x) ={5x+5−51​x+56​​ if x≤1 if x>1​

Determine whether the function is an even function, an odd function, or neither. - g(x) =ex3−4g ( x ) = e ^ { x ^ { 3 } } - 4g(x) =ex3−4

Graph the greatest integer function. - $f ( x ) = \llbracket \frac { 1 } { 3 } x \rrbracket$ $Graph the greatest integer function. - f ( x ) = \llbracket \frac { 1 } { 3 } x \rrbracket A) B) C) D)$

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - $f ( x ) = x ^ { 2 } - 25$

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - $f ( x ) = x ^ { 2 } - 16 x + 64$

Determine whether the function is an even function, an odd function, or neither. - $f ( x ) = x ^ { 5 } - x ^ { 4 }$

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - $f ( x ) = x ^ { 3 } + x ^ { 2 } - 20 x$

Determine whether the function is an even function, an odd function, or neither. - $f ( x ) = x ^ { 3 } + x ^ { 2 } - 4$

Determine whether the function is an even function, an odd function, or neither. - $f ( x ) = x ^ { 3 } - 4 x$

Determine whether the function is an even function, an odd function, or neither. - $g ( x ) = e ^ { 3 x } + 5$

Graph the greatest integer function. - $f ( x ) = \llbracket \frac { x + 4 } { 3 } \rrbracket$ $Graph the greatest integer function. - f ( x ) = \llbracket \frac { x + 4 } { 3 } \rrbracket A) B) C) D)$

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - $f ( x ) = 5 ( x + 3 ) ( x + 7 ) ^ { 4 }$

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - $f ( x ) = 2 x ^ { 2 } - 26$

Graph the greatest integer function. - $f ( x ) = \llbracket x - 2 \rrbracket$ $Graph the greatest integer function. - f ( x ) = \llbracket x - 2 \rrbracket A) B) C) D)$

Find the real zeros of the function, state their multiplicities if it is a number other than 1. - $f ( x ) = x ^ { 2 } + 10 x + 21$

Determine whether the function is an even function, an odd function, or neither. - $f ( x ) = 2 x ^ { 2 } + x ^ { 4 }$

Determine whether the function is an even function, an odd function, or neither. - $f ( x ) = - 3 x ^ { 5 } + x ^ { 3 }$

Determine whether the function is an even function, an odd function, or neither. - $g ( x ) = e ^ { x ^ { 3 } } - 4$