Functions and function notation

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A restaurant sets the prices of its dishes using the following function:

Price = (Cost of Ingredients) + (40% of the Cost of Ingredients) + 5

where all quantities are in U.S. Dollars.

If the cost of ingredients for a steak dish is $14, what will the restaurant set the price of the steak dish at?

$24.60

$19.60

$19

$13.40

$14

Explanation

The price of the dish is a function of a single variable, the cost of the ingredients. One way to conceptualize the problem is by thinking of it in function notation. Let be the variable representing the cost of the ingredients. Let be a function of the cost of ingredients giving the price of the dish. Then, we can turn

"Price = (Cost of Ingredients) + (40% of the Cost of Ingredients) + 5"

into a regular equation with recognizable parts. Replace "Price" with and replace "cost of ingredients" with the variable .

Simplify by combining like terms ( and ) to obtain:

The cost of ingredients for the steak dish is $14, so substitute 14 for .

$p(14) = 1.4\times14 +5$

All that's left is to compute the answer:

So, the steak dish will have a price of $24.60.

A restaurant sets the prices of its dishes using the following function:

Price = (Cost of Ingredients) + (40% of the Cost of Ingredients) + 5

where all quantities are in U.S. Dollars.

If the cost of ingredients for a steak dish is $14, what will the restaurant set the price of the steak dish at?

$24.60

$19.60

$19

$13.40

$14

Explanation

"Price = (Cost of Ingredients) + (40% of the Cost of Ingredients) + 5"

into a regular equation with recognizable parts. Replace "Price" with and replace "cost of ingredients" with the variable .

Simplify by combining like terms ( and ) to obtain:

The cost of ingredients for the steak dish is $14, so substitute 14 for .

$p(14) = 1.4\times14 +5$

All that's left is to compute the answer:

So, the steak dish will have a price of $24.60.

Consider the scenario below:

Helen is a painter. It takes her 3 days to make each painting. She has already made 6 paintings. Which of the following functions best models the number of paintings she will have after days?

$f(x)=\frac{x}{3} +6$

Explanation

The question asks, "Which of the following functions best models the number of paintings she will have after days?"

From this, you know that the variable represents the number of days, and that represents the number of paintings she makes as a function of days spent working.

If it takes 3 days to make a painting, each day results in $\frac{1}{3}$ paintings. Therefore, we have a linear relationship with slope $\frac{1}{3}$ .

Additionally, she begins with 6 paintings. Therefore, even when zero days are spent working on paintings, she will have 6 paintings. In other words, . This means the y-intercept is 6.

As a result, the function will be

$f(x)=\frac{1}{3}x +6$

which can be rewritten as

$f(x)=\frac{x}{3} +6$

Define

$f(x)= \left\{\begin{matrix} x-2,&x< 0 \ x+ 3,& x \ge 0\end{matrix}\right.$

Give the range of the function.

$(-\infty, -2) \cup [3, \infty)$

$(-\infty, \infty)$

$(-\infty, -2] \cup (3, \infty)$

Explanation

The range of a function is the set of all possible values of over its domain.

Since this function is piecewise-defined, it is necessary to examine both parts of the function and extract the range of each.

For , it holds that , so

$x \in(-\infty, -2)$ .

For $x \ge 0$ , it holds that , so

$x \ge 0$

$x+ 3 \ge 0+ 3$

$f(x) \ge 3$ ,

$x \in [3, \infty)$

The overall range of is the union of these sets, or $(-\infty, -2) \cup [3, \infty)$ .

Consider the scenario below:

Helen is a painter. It takes her 3 days to make each painting. She has already made 6 paintings. Which of the following functions best models the number of paintings she will have after days?

$f(x)=\frac{x}{3} +6$

Explanation

The question asks, "Which of the following functions best models the number of paintings she will have after days?"

From this, you know that the variable represents the number of days, and that represents the number of paintings she makes as a function of days spent working.

If it takes 3 days to make a painting, each day results in $\frac{1}{3}$ paintings. Therefore, we have a linear relationship with slope $\frac{1}{3}$ .

Additionally, she begins with 6 paintings. Therefore, even when zero days are spent working on paintings, she will have 6 paintings. In other words, . This means the y-intercept is 6.

As a result, the function will be

$f(x)=\frac{1}{3}x +6$

which can be rewritten as

$f(x)=\frac{x}{3} +6$

Define

$f(x)= \left\{\begin{matrix} x-2,&x< 0 \ x+ 3,& x \ge 0\end{matrix}\right.$

Give the range of the function.

$(-\infty, -2) \cup [3, \infty)$

$(-\infty, \infty)$

$(-\infty, -2] \cup (3, \infty)$

Explanation

The range of a function is the set of all possible values of over its domain.

Since this function is piecewise-defined, it is necessary to examine both parts of the function and extract the range of each.

For , it holds that , so

$x \in(-\infty, -2)$ .

For $x \ge 0$ , it holds that , so

$x \ge 0$

$x+ 3 \ge 0+ 3$

$f(x) \ge 3$ ,

$x \in [3, \infty)$

The overall range of is the union of these sets, or $(-\infty, -2) \cup [3, \infty)$ .

Define functions $f(x) = \sqrt{x-8}$ and $g(x) = \sqrt{x+ 8}$ .

Give the domain of the function .

$[8, \infty )$

$[-8, \infty )$

$(-\infty , 8]$

$(-\infty , -8]$

Explanation

The domain of the sum of two functions is the intersections of the domains of the individual functions. Both and are square root functions, so their radicands must both be positive.

The domain of is the set of all values of such that:

$x-8 \ge 0$

Adding 8 to both sides:

$x-8 + 8 \ge 0 + 8$

$x \ge 8$

This domain is $[8, \infty )$ .

The domain of is the set of all values of such that:

$x+8 \ge 0$

Subtracting 8 tfrom both sides:

$x+ 8-8 \ge 0 - 8$

$x \ge -8$

This domain is $[-8, \infty )$

The domain of is the intersection of the domains, which is $[8, \infty )$ .

Define functions $f(x) = \sqrt{x-8}$ and $g(x) = \sqrt{x+ 8}$ .

Give the domain of the function .

$[8, \infty )$

$[-8, \infty )$

$(-\infty , 8]$

$(-\infty , -8]$

Explanation

The domain of the sum of two functions is the intersections of the domains of the individual functions. Both and are square root functions, so their radicands must both be positive.

The domain of is the set of all values of such that:

$x-8 \ge 0$

Adding 8 to both sides:

$x-8 + 8 \ge 0 + 8$

$x \ge 8$

This domain is $[8, \infty )$ .

The domain of is the set of all values of such that:

$x+8 \ge 0$

Subtracting 8 tfrom both sides:

$x+ 8-8 \ge 0 - 8$

$x \ge -8$

This domain is $[-8, \infty )$

The domain of is the intersection of the domains, which is $[8, \infty )$ .

The daily pay in U.S. Dollars for a certain job is defined as the following function , where equals time in hours:

If an employee works for 7 hours in a day, how much is he or she paid?

$\$115.00$

$\$105.00$

$\$70.00$

$\$85.00$

$\$25.00$

Explanation

The above function can be understood as, "An employee is paid $15 for each hour he or she works, plus a flat amount of $10."

If an employee works 7 hours, we can find the amount that he or she is paid by plugging in 7 for "Hours" in the equation:

Adhering to order of operations, we next find the product of 15 and 7:

Finally, we find the sum of 105 and 10:

The employee is paid $115.00 for seven hours' work.

Define functions $f(x) = \frac{1}{x-1}$ and $g(x)= \sqrt{x}+ 1$ .

Give the domain of the composition $g\circ f$ .

$(1, \infty)$

$(0, 1) \cup (1, \infty)$

$(-\infty, 1)$

$(-\infty, 1) \cup (1, \infty)$

Explanation

The domain of the composition of functions $g\circ f$ is that the set of all values that fall in the domain of such that falls in the domain of .

$f(x) = \frac{1}{x-1}$ , a rational function, so its domain is the set of all numbers except those that make its denominator zero. if and only if , making its domain $(-\infty, 1) \cup (1, \infty)$ .

$g(x)= \sqrt{x}+ 1$ , a square root function, so the domain of is the set of all values that make the radicand a nonnegative number. Since the radicand is itself, the domain of is simply the set of all nonnegative numbers, or $\left [ 0, \infty\right )$ .