$\textsf{$f(1) = \boxed{1}$ , meaning when a $\boxed{1}$ is rolled on the die, the player is awarded}\\\\ \textsf{ $\boxed{1}$ point. This interpretation $\boxed{\sf makes \; sense}$ in the context of the problem.}$

$\textsf{$f(4.5) = \boxed{8}$ , meaning when a $\boxed{4.5}$ is rolled on the die, the player is awarded} \\\\ \textsf{ $\boxed{8}$ points. This interpretation $\boxed{\sf does \; not \; make \; sense}$ in the context of the problem.}$

$\textsf{$f(10) = \boxed{19}$ , meaning when a $\boxed{10}$ is rolled on the die, the player is awarded} \\\\ \textsf{ $\boxed{19}$ points. This interpretation $\boxed{\sf does \; not \; make \; sense}$ in the context of the problem.}$

$\textsf{Based on the observations above, it is clear that an appropriate domain for the}\\\\ \textsf{function is $\boxed{ \{1, 2, 3, 4, 5, 6\}}$ .}$

Step-by-step explanation:

<u>Given function</u>:

$f(x)=2x-1$

where:

x is the value rolled on the six-sided die.
The sides of the die are labelled 1 to 6.

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f(1) means the value of the function when x = 1.

Therefore, substitute x = 1 into the given function to find f(1):

$\begin{aligned}x=1 \implies f(1)&=2(1)-1\\&=2-1\\&=1\end{aligned}$

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f(4.5) means the value of the function when x = 4.5.

Therefore, substitute x = 4.5 into the given function to find f(4.5):

$\begin{aligned}x=4.5 \implies f(4.5)&=2(4.5)-1\\&=9-1\\&=8\end{aligned}$

As the faces of the six-sided die are labelled 1 to 6, the only values of x that make sense are 1, 2, 3, 4, 5 and 6. Therefore, rolling a "4.5" does not make sense.

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f(10) means the value of the function when x = 10.

Therefore, substitute x = 10 into the given function to find f(10):

$\begin{aligned}x=10 \implies f(1)&=2(10)-1\\&=20-1\\&=19\end{aligned}$

As the faces of the six-sided die are labelled 1 to 6, the only values of x that make sense are 1, 2, 3, 4, 5 and 6. Therefore, rolling a "10" does not make sense.

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The domain of a function is the set of all possible x-values.

As the faces of the six-sided die are labelled 1 to 6, the only possible values of x are 1, 2, 3, 4, 5 and 6.

$\textsf{Based on the observations above, it is clear that an appropriate domain for the}\\\\ \textsf{function is $\boxed{ \{1, 2, 3, 4, 5, 6\}}$ .}$

The domain can also be written as:

$\{x \in \mathbb{N} \; | \; 1 \leq x \leq 6 \}$

or $\{x \in \mathbb{Z} \; | \; 1 \leq x \leq 6 \}$

8 0

1 year ago

NEED HELP WITH MY MATH

nataly862011 [7]

So for this problem we can recognize that it's a system of linear equations

recall the slope intercept form of a line
$y = mx + b$
given the info we can write two linear equations one for John and one for Lyndon

take a look at johns part of the question. we can see that his bank starts with 100 dollars and every day after he saves 20 dollars. Now we can see that the 100 is the y intercept or b of our equation because it's our initial amount and the rate of change or m of the equation is 25. filling in this info we get this as johns equation:

$y = 25x + 100$
now we can do the same for Lydon, she has 50 dollars to start and saves 30 dollars a day. So using the same steps as we did for John we get the equation
$y = 30x + 50$
now we can do a little trick, to find out when their bank amounts will be equal we set both equations equal to each other (we're just substituting y into the equation) So we get
$25x + 100 = 30x + 50$
now let's clean this up and solve for X with some algebra
$100 = 30x - 25x + 50$
subtract 25x
$100 = 5x + 20$
combine like terms
$80 = 5x$
subtract 20
$x = 16$
divide off 5 to get your x value

now we're almost done we just need to plug X back into any of our two equations we made at the beginning of the problem lets use johns equation and plug in 16 because as we found out x=16

$y = 25(16) + 100$
multiply and add to get
$y = 500$

now that we know
$x = 16 \: \: and \: \: y = 500$
we can finalize our answer answer to the system as a coordinate (x,y) and we'll get

(16,500)

since the questions asks after how many weeks though we would say 16 or the x value because in a time related problem, time is always the x value so the final answer is

$16 \: \: weeks$

3 0

3 years ago