Math / Algebra

QuestionA hardware store rents vacuum cleaners that customers may use for part of a day before returning. The function $f(x)=6 x+14$ models the total rental cost of a vacuum cleaner.
Which is a reasonable domain for the function? A. $14<x<32$ B. $0<x<25$ C. $0<x<12$ D. $14<x<86$

Studdy Solution

STEP 1

What is this asking? What are the reasonable values for $x$ (part of a day) given the total rental cost, $f(x)$ , of a vacuum cleaner? Watch out! Don't mix up the domain (input values of $x$ ) and the range (output values of $f(x)$ ).
We're looking for reasonable values of $x$ , not $f(x)$ !

STEP 2

1. Understand the function
2. Analyze the answer choices

STEP 3

Alright, so we've got this function $f(x) = 6x + 14$ , where $f(x)$ is the total rental cost in $\$$ and $x$ represents the fraction of a day the vacuum cleaner is rented for.

STEP 4

Let's break it down!
The $14$ represents a fixed cost, like a base fee just for renting the vacuum cleaner, regardless of how long you use it.
The $6x$ part represents a variable cost that depends on how long you rent it for.
The $6$ is the rate of change, meaning it costs $\$6$ for each additional fraction of a day you rent the vacuum.

STEP 5

Since $x$ represents a fraction of a day, it must be between $0$ and $1$ .
If $x=0$ , that means you didn't rent it at all, and if $x=1$ you rented it for the whole day.
Any value between $0$ and $1$ represents a fraction of a day.

STEP 6

Let's look at the answer choices and see which one makes sense for our fraction of a day, $x$ .

STEP 7

Answer choice A, $14 < x < 32$ , doesn't make sense because $x$ can't be greater than $1$ .
Remember, $x$ is a fraction of a day.

STEP 8

Answer choice B, $0 < x < 25$ , also doesn't work because $x$ can't be greater than $1$ .

STEP 9

Answer choice C, $0 < x < 12$ , is closer, but still not quite right.
While the lower bound of $0$ is correct, the upper bound of $12$ is too high. $x$ can't be greater than $1$ .

STEP 10

Answer choice D, $14 < x < 86$ , has the same issue – both bounds are way too high!

STEP 11

Since none of the given options are correct, let's determine a reasonable domain.
We know $x$ should be between $0$ and $1$ inclusive, so a reasonable domain would be $0 \le x \le 1$ .
If we must choose from the given options, C is the closest, but we need to adjust it to $0 < x \le 1$ .
If we can't rent for a full day ( $x=1$ ), then $0 < x < 1$ would be the most accurate representation.

STEP 12

The most reasonable domain for this function, given that $x$ represents a fraction of a day, is $0 < x \le 1$ if we can rent for a full day, or $0 < x < 1$ if we can only rent for part of a day.
Since we have to choose, and option C is the closest, we'll choose C and keep in mind it should really be $0 < x < 1$ .

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QuestionA hardware store rents vacuum cleaners that customers may use for part of a day before returning. The function $f(x)=6 x+14$ models the total rental cost of a vacuum cleaner.
Which is a reasonable domain for the function? A. $14<x<32$ B. $0<x<25$ C. $0<x<12$ D. $14<x<86$

Studdy Solution

STEP 1

What is this asking? What are the reasonable values for $x$ (part of a day) given the total rental cost, $f(x)$ , of a vacuum cleaner? Watch out! Don't mix up the domain (input values of $x$ ) and the range (output values of $f(x)$ ).
We're looking for reasonable values of $x$ , not $f(x)$ !

STEP 2

1. Understand the function
2. Analyze the answer choices

STEP 3

Alright, so we've got this function $f(x) = 6x + 14$ , where $f(x)$ is the total rental cost in $\$$ and $x$ represents the fraction of a day the vacuum cleaner is rented for.

STEP 4

STEP 5

Since $x$ represents a fraction of a day, it must be between $0$ and $1$ .
If $x=0$ , that means you didn't rent it at all, and if $x=1$ you rented it for the whole day.
Any value between $0$ and $1$ represents a fraction of a day.

STEP 6

Let's look at the answer choices and see which one makes sense for our fraction of a day, $x$ .

STEP 7

Answer choice A, $14 < x < 32$ , doesn't make sense because $x$ can't be greater than $1$ .
Remember, $x$ is a fraction of a day.

STEP 8

Answer choice B, $0 < x < 25$ , also doesn't work because $x$ can't be greater than $1$ .

STEP 9

Answer choice C, $0 < x < 12$ , is closer, but still not quite right.
While the lower bound of $0$ is correct, the upper bound of $12$ is too high. $x$ can't be greater than $1$ .

STEP 10

Answer choice D, $14 < x < 86$ , has the same issue – both bounds are way too high!

STEP 11

STEP 12

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Studdy Solution

STEP 1

STEP 2

1. Understand the function2. Analyze the answer choices

STEP 3

Alright, so we've got this function f(x)=6x+14f(x) = 6x + 14f(x)=6x+14, where f(x)f(x)f(x) is the **total rental cost** in $\$$ and xxx represents the **fraction of a day** the vacuum cleaner is rented for.

STEP 4

STEP 5

Since xxx represents a fraction of a day, it must be between 000 and 111.If x=0x=0x=0, that means you didn't rent it at all, and if x=1x=1x=1 you rented it for the whole day.Any value between 000 and 111 represents a fraction of a day.

STEP 6

Let's look at the answer choices and see which one makes sense for our fraction of a day, xxx.

STEP 7

Answer choice A, 14<x<3214 < x < 3214<x<32, doesn't make sense because xxx can't be greater than 111.Remember, xxx is a **fraction of a day**.

STEP 8

Answer choice B, 0<x<250 < x < 250<x<25, also doesn't work because xxx can't be greater than 111.

STEP 9

Answer choice C, 0<x<120 < x < 120<x<12, is closer, but still not quite right.While the lower bound of 000 is correct, the upper bound of 121212 is too high. xxx can't be greater than 111.

STEP 10

Answer choice D, 14<x<8614 < x < 8614<x<86, has the same issue – both bounds are way too high!

STEP 11

STEP 12

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1. Understand the function
2. Analyze the answer choices

Alright, so we've got this function $f(x) = 6x + 14$ , where $f(x)$ is the total rental cost in $\$$ and $x$ represents the fraction of a day the vacuum cleaner is rented for.

Since $x$ represents a fraction of a day, it must be between $0$ and $1$ .
If $x=0$ , that means you didn't rent it at all, and if $x=1$ you rented it for the whole day.
Any value between $0$ and $1$ represents a fraction of a day.

Let's look at the answer choices and see which one makes sense for our fraction of a day, $x$ .

Answer choice A, $14 < x < 32$ , doesn't make sense because $x$ can't be greater than $1$ .
Remember, $x$ is a fraction of a day.

Answer choice B, $0 < x < 25$ , also doesn't work because $x$ can't be greater than $1$ .

Answer choice C, $0 < x < 12$ , is closer, but still not quite right.
While the lower bound of $0$ is correct, the upper bound of $12$ is too high. $x$ can't be greater than $1$ .

Answer choice D, $14 < x < 86$ , has the same issue – both bounds are way too high!