Mathematics, Functions, Linear Functions

by LII
(United States)

Can You please help me with these 3 questions?

Which of the following are functions? The last two problems, i.e., b & c, are multi part relations consider all parts when determining whether or not these relations are functions. Explain your reasoning for a, b, and c.

a. f(x) = x + 3

b. f(x) = 73 if x>2 otherwise f(x) = -1

c. f(x) = 79if x>0 or f(x) = -9 if x<0 or f(x) = 9 or -9 if x = 0

Suppose you have a lemonade stand, and when you charge $1 per cup of lemonade you sell 48 cups. But when you raise your price to $2 you only sell 27 cups. Write an equation for the number of cups you sell as a function of the price you charge. Denote "C" for number of cups, and "P" for the price you charge. Assume the function is linear. Show your work.

Take a look at the table below and write out an equation for f(x). Show your work.

x -3 -2 1 3 4
f(x) 0 3 12 18 21

For each of the relationships below, explain whether you think it is best described by a linear function or a non-linear function. Explain your reasoning thoroughly

a. A person's height as a function of the person's age (from age 0 to 100)

b. The probability of getting into a car accident as a function of the speed at which you drive

c. The time it takes you to get to work as a function the speed at which you drive

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Feb 04, 2011

Mathematics, Functions, Linear Functions
by: Staff

-------------------------------------------------------------------------------

PART III

Question 3

Take a look at the table below and write out an equation for f(x). Show your work.

x -3 -2 1 3 4
f(x) 0 3 12 18 21

1. Determine whether this relationship is a linear relationship by computing the slope between the data points.

Slope = rise/run

Slope = (y2 – y1)/(x2 – x1)

Slope from x = -3 to x = -2

Slope = (3 – 0)/[-2 – (-3)]

Slope = (3)/[1]

Slope = 3

Slope from x = -2 to x = 1

Slope = (12 – 3)/[1 – (-2)]

Slope = (9)/[3]

Slope = 3

Slope from x = 1 to x = 3

Slope = (18 – 12)/[3 – (1)]

Slope = (6)/[2]

Slope = 3

Slope from x = 3 to x = 4

Slope = (21 – 18)/[4 – (3)]

Slope = (3)/[1]

Slope = 3

The relationship is a linear relationship. In all three cases the slope = 3.

Since this is the case, we can apply the slope intercept form of a linear equation.

y = mx + b

compute m, the slope of the function, by using any 2 data points. This is exactly the same technique we used to solve question 2.

For example: (1,12) & (3,18)

12 = m*1 + b

18 = m*3 + b

12 = m*1 + b
-(18 = m*3 + b)
--------------------

-6 = -2m + 0

-6 = -2m

(-6)/(-2) = (-2m)/(-2)

3 = m (this is the same slope we computed between the data points – this computation serves as a double check)

FIND the SLOPE INTERCEPT b.

y = mx + b

y = 3*x + b

12 = 3*1 + b

12 = 3 + b

12 - 3 = 3 - 3 + b

9 = 0 + b

9 = b

The final equation for question 3 is:

y = mx + b

y = 3x + 9

the final answer to question 3: f(x) = 3x + 9

Question 4

For each of the relationships below, explain whether you think it is best described by a linear function or a non-linear function. Explain your reasoning thoroughly

a. A person's height as a function of the person's age (from age 0 to 100)

This is a non-linear function. A person’s height is very small when they are born. As time passes, a person grows to a maximum height. As a person grows old, their height decreases somewhat.

b. The probability of getting into a car accident as a function of the speed at which you drive.

This is also a non-linear function.

Click the link below to see how driving speed affects the probability of getting into an accident:

http://ec.europa.eu/transport/wcm/road_safety/erso/knowledge/Content/20_speed/speed_and_accident_risk.htm

c. The time it takes you to get to work as a function the speed at which you drive

This is a linear function.

Time = distance/speed

Thanks for writing.

Staff
www.solving-math-problems.com

Feb 04, 2011

Mathematics, Functions, Linear Functions
by: Staff

-------------------------------------------------------------------------------

PART II

Question 2

Suppose you have a lemonade stand, and when you charge $1 per cup of lemonade you sell 48 cups. But when you raise your price to $2 you only sell 27 cups. Write an equation for the number of cups you sell as a function of the price you charge. Denote "C" for number of cups, and "P" for the price you charge. Assume the function is linear. Show your work.

You know the FUNCTION IS LINEAR from the statement of the problem.

You also know TWO DATA POINTS:

$1 per cup, 48 cups

$2 per cup, 27 cups

To write an equation, apply the slope intercept form of a linear equation:

y = mx + b

substituting the variables names you would like to use:

C = m*P + b

(m is the slope of the equation. b is the slope intercept.)

FIND the SLOPE m.

Data Point 1: P = $1 per cup, C = 48 cups

48 = m*1 + b

Data Point 2: P = $2 per cup, C = 27 cups

27 = m*2 + b

You now have two equations with two variables.

48 = m*1 + b
27 = m*2 + b

Subtract the second equation from the first equation to eliminate the b

48 = m*1 + b
-(27 = m*2 + b)

Applying the distributive law

48 = m*1 + b
-27 = -m*2 - b

After the subtraction

21 = -m + 0

21 = -m

To reverse the signs, multiply each side of the equation by -1.

21 = -m

(-1)*21 = (-1)*(-m)

-21 = m

m = -21

The slope (m) of your equation is -21. Since the slope m is negative, a graph of the equation will show a straight line sloping downward (left to right).

Since we know the value of m, your equation now looks like this:

C = m*P + b

C = (-21)*P + b

To complete the problem, we still need to find b, the slope intercept.

FIND the SLOPE INTERCEPT b.

C = (-21)*P + b

Use either one of the data points to find b. I will use data point 1.

C = (-21)*P + b

P = $1 per cup, C = 48 cups

Substitute $1 per cup for the P, and 48 cups for the C

C = (-21)*P + b

48 = (-21)*1 + b

48 = -21 + b

Solve for b

Add 21 to each side of the equation:

48 + 21 = -21 + 21 + b

69 = -21 + 21 + b

69 = 0 + b

69 = b

b = 69

the slope intercept, b = 69

The final equation is:

m = -21, b = 69

C = m*P + b

C = -21*P + 69

The final answer to question 2 is: C = -21*P + 69

If you plot this function, the result will be as follows (click link to view):

http://www.solving-math-problems.com/images/Linear-Function-lemonade-stand.png

Feb 04, 2011

Mathematics, Functions, Linear Functions
by: Staff

The question:

by LII
(United States)

Can You please help me with these 3 questions?

Which of the following are functions? The last two problems, i.e., b & c, are multi part relations consider all parts when determining whether or not these relations are functions. Explain your reasoning for a, b, and c.

a. f(x) = x + 3

b. f(x) = 73 if x>2 otherwise f(x) = -1

c. f(x) = 79if x>0 or f(x) = -9 if x<0 or f(x) = 9 or -9 if x = 0

Suppose you have a lemonade stand, and when you charge $1 per cup of lemonade you sell 48 cups. But when you raise your price to $2 you only sell 27 cups. Write an equation for the number of cups you sell as a function of the price you charge. Denote "C" for number of cups, and "P" for the price you charge. Assume the function is linear. Show your work.

Take a look at the table below and write out an equation for f(x). Show your work.

x -3 -2 1 3 4
f(x) 0 3 12 18 21

For each of the relationships below, explain whether you think it is best described by a linear function or a non-linear function. Explain your reasoning thoroughly

a. A person's height as a function of the person's age (from age 0 to 100)

b. The probability of getting into a car accident as a function of the speed at which you drive

c. The time it takes you to get to work as a function the speed at which you drive

The answer:

PART I

Question 1

Which of the following are functions? The last two problems, i.e., b & c, are multi part relations consider all parts when determining whether or not these relations are functions. Explain your reasoning for a, b, and c.

a. f(x) = x + 3

b. f(x) = 73 if x>2 otherwise f(x) = -1

c. f(x) = 79if x>0 or f(x) = -9 if x<0 or f(x) = 9 or -9 if x = 0

Not all relationships are functions. Not all equations are functions. A function is a special type of relationship.

For any of your choices to be a function, every x MUST CORRESPOND TO ONLY ONE f(x) – ONLY ONE.

Choice a: f(x) = x + 3. CHOICE “A” IS A FUNCTION.

For example, if x = 1, then f(1) = 4 . . . ONLY 4 . . . not 4 and some other number at the same time. If x = 5, then f(5) = 8 . . . ONLY 8 . . . not 8 and some other number at the same time . . . and so on.

Choice b: f(x) = 73 if x>2 otherwise f(x) = -1. CHOICE “B” IS ALSO A FUNCTION.

For example, if x = 1, then f(1) = -1 . . . ONLY -1 . . . not -1 and some other number at the same time. If x = 2, then f(2) = -1 . . . ONLY -1 . . . not -1 and some other number at the same time. If x = 3, then f(3) = 73 . . . ONLY 73 . . . not 73 and some other number at the same time.

Choice c: f(x) = 79 if x>0 or f(x) = -9 if x<0 or f(x) = 9 or -9 if x = 0. CHOICE “C” IS NOT A FUNCTION.

If x = 0, then f(0) = 9 or -9 . . . TWO NUMBERS AT THE SAME TIME.

The final answer for Question 1: choices a and b are functions. Choice c is not a function.

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Mathematics, Functions, Linear Functions

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