EOC Functions & Modeling

EOC Functions & Modeling

by Darryl Padgett

What is a composite function?

• Can be thought of a series of operations

• One function is substituted into another function

• Often denoted as f(g(x)) or (f ∘ g)(x)

• Read as "f of g of x"

• It is an operation on functions

• You are multiplying f(x) and g(x) not substituting functions

Composite Functions Example

Multiple Choice

Which expression represents g(f(x)) if f(x) = 2x - 8 and g(x) = 4x

• 8x - 32
• 8x² - 32x
• 8x - 8
• 6x - 8

Multiple Choice

If f(x) = x² and g(x) = 3x - 1, find f(g(x))

• 9x² - 6x + 1

• 3x - 1

• 3x² - 1

• 9x² - 1

Multiple Choice

If f(x) = -4x - 5 and
g(x) = 3 - x,
what is g(-4) + f(1)?

• 7
• -2
• -9
• -10

Multiple Choice

f(x) = 3x + 10
g(x) = x - 2
Find g(f(-10))

• -42
• -26
• -18
• None of these.

Multiple Choice

In 2014, the cost to mail a letter was 49¢ for up to one ounce. Every additional ounce cost 21¢. Which recursive function could be used to determine the cost of a 3-ounce letter, in cents? 

• 𝑎₁ = 49;

  𝑎_𝑛 = 𝑎_𝑛−1 + 21 

• 𝑎₁ = 0;

  𝑎_𝑛 = 49(𝑎_{𝑛 − 1} ) + 21 

• 𝑎₁ = 21;

  𝑎_𝑛 = 𝑎_𝑛−1 + 49 

• 𝑎₁ = 0;

  𝑎_𝑛 = 21(𝑎_𝑛−1) + 49 

Multiple Choice

Describe the transformation that occurred.

h(x) = f(x) + 2

• up 2 units

• left 2 units

• Vertical Strech

• right 2 units

Multiple Choice

Given f(x) = (x-3)² + 5.  What transformations took place from the original function f(x)?

• Left 3 and up 5
• Right 3 and down 5
• Left 3 and down 5
• Right 3 and up 5

Multiple Choice

Match the equation to its description.

• Right 2 and up 2
• Left 2 and up 2
• Right 2 and down 2
• Left 2 and down 2

Multiple Choice

Describe the transformation from dotted to solid.

• f(x + 4)
• f(x - 4)
• f(x) + 4
• f(x) - 4

Multiple Choice

If the blue is f(x)=x^2, the red must be

• g(x) = (x+3)²

• g(x) = (x-3)²

• g(x) = x²+3

• g(x) = x²-2

Multiple Choice

-f(x) - 2

• reflection over y-axis, shift down 2

• reflection over y-axis, shift right 2

• reflection over x-axis, shift right 2

• reflection over x-axis, shift down 2

Multiple Choice

The height, ℎ, in feet, of an object thrown upward from a height of 144 feet is a function of time, 𝑡, in seconds. The height can be determined by the function ℎ(𝑡)=−16𝑡² + 128𝑡 + 144 . What is the height of the object at 3 seconds? 

• 144

• 384

• 432

• 672

Multiple Choice

The number of miles a car can be driven depends on the number of gallons of gas in its tank. The function m = 25g models a situation in which a car gets 25 miles per gallon. If the gas tank holds 20 gallons of gas, which inequality represents its range? 

• 0 ≤ 𝑔 ≤ 20 

• 0 ≤ 𝑚 ≤ 500 

• 𝑚 ≤ 500 

• 𝑔 ≤ 20 

Multiple Choice

Let 𝑓 be a function such that 𝑓(𝑥) = 2𝑥 − 4 is defined on the domain 2 ≤ 𝑥 ≤ 6. The range of this function is 

• −∞ ≤ 𝑦 ≤ ∞ 

• 0 ≤ 𝑦 ≤ 8 

• 0 ≤ 𝑦 ≤ ∞ 

• 2 ≤ 𝑦 ≤ 6 

Multiple Choice

Given that 𝑦 is a function of 𝑥, which of the following tables best represents a function?  Hint: Remember that each input must have a different output.

Multiple Select

The domain of the function 𝑓(𝑥) = −3𝑥 is restricted to the negative integers. (The domain is the x values). Which values are elements of the range? 

• -12

• -3

• 12

• 7

• 9

Multiple Choice

Sue hits a ball from a height of 4 feet. The height of the ball above the ground is a function of the horizontal distance the ball travels until it comes to rest on the ground. What is the maximum height that the ball travels?

• 0 feet

• 50 feet

• 100 feet

• 200feet

Fill in the Blanks

James has a cell phone plan that includes 200 minutes for phone calls and unlimited texting. An additional fee is charged for using more than 200 minutes for phone calls. The figure below is the graph of 𝐶 = 𝑓(𝑚), where 𝐶 is the monthly cost after m minutes used.  What is the minimum monthly cost for Corinne's cell phone plan?  You do not have to add the $ sign.

Fill in the Blanks

James has a cell phone plan that includes 200 minutes for phone calls and unlimited texting. An additional fee is charged for using more than 200 minutes for phone calls. The figure below is the graph of 𝐶 = 𝑓(𝑚), where 𝐶 is the monthly cost after m minutes used. 

For what 𝑚 is 𝑓(𝑚) = 55?   (Enter number only)

Fill in the Blanks

James has a cell phone plan that includes 200 minutes for phone calls and unlimited texting. An additional fee is charged for using more than 200 minutes for phone calls. The figure below is the graph of 𝐶 = 𝑓(𝑚), where 𝐶 is the monthly cost after m minutes used.  What is the cost per minute after Corinne uses her monthly allowance of 200 minutes?  Hint! find the slope of the line. (Enter the number only)

Fill in the Blanks

James has a cell phone plan that includes 200 minutes for phone calls and unlimited texting. An additional fee is charged for using more than 200 minutes for phone calls. The figure below is the graph of 𝐶 = 𝑓(𝑚), where 𝐶 is the monthly cost after m minutes used.  What is the value of 𝑓(300)?   You do not have to add the $ sign.

Multiple Select

The function 𝑓(𝑥) = 4𝑥− 𝑥² is graphed in the

𝑥𝑦-coordinate plane as shown. 

Based on the graph of the function, which statements are true? Select ALL that apply. 

• f is increasing on the interval

  0 < 𝑥 < 2. 

• f is decreasing on the interval

  0 < 𝑥 < 2. 

• f is increasing on the interval

  2 < 𝑥 < 4. 

• f is decreasing on the interval

  2 < 𝑥 < 4. 

• f is increasing on the interval 𝑥 > 4. 

Multiple Select

The function 𝑓(𝑥) = 4𝑥− 𝑥² is graphed in the

𝑥𝑦-coordinate plane as shown. 

Based on the graph of the function, which statements are true? Select ALL that apply. 

• f(x) < 0 on the interval 𝑥 < 0. 

• f(x) < 0 on the interval 0 < 𝑥 < 2 . 

• f(x) $$\ge$$ 0 on the interval 0 < 𝑥 < 2 . 

• f(x) > 0 on the interval 2 < 𝑥 < 4 . 

• f(x) < 0 on the interval 𝑥 > 4 . 

Multiple Choice

A computer technician charges a one-time fee of $50 plus an additional $20 per hour of labor. If an equation is created to determine the technician's total charge, what does the $50 represent?  Hint: create an equation of the line.

• slope

• coefficient

• x-intercept

• y-intercept

Multiple Choice

What is the maximum height of the fire hose water?

• 4 feet

• 24 feet

• 41 feet

• 45 feet

Fill in the Blanks

The water spray will reach the ground

feet from the base of the fire hose.  Enter the number only

Fill in the Blanks

A grasshopper jumps off of a tree stump. The height, in feet, of the grasshopper above the ground after t seconds is modeled by the function shown. 

ℎ(𝑡) = −𝑡² + 43𝑡 + 0.25. After how many seconds will the grasshopper land on the ground? (Hint use Desmos to graph the function). Enter number only!

Multiple Choice

The distance in miles, 𝑦, a bicyclist is from home after riding 𝑥 hours is represented by the equation 𝑦 = 8𝑥 + 7. What does the slope represent in this situation? 

• the number of hours it takes the bicyclist to ride 15 miles 

• the distance the bicyclist is from home when 𝑥 = 0 

• the steepness of the hill the bicyclist is climbing 

• the speed of the bicyclist 

Multiple Choice

Which is the graph of a linear function with a slope of 2 and a y-intercept at (0, 1)? 

Multiple Choice

What is the equation of the function represented by this table of values? 

• 𝑦 = 5𝑥+3 

• 𝑦 = 12𝑥+3 

• 𝑦 = 3∙5^𝑥 

• 𝑦 = 5∙3^𝑥 

Multiple Choice

Which expression represents the output of the n^th term? 

• 𝑛 + 2 

• 𝑛 + 11 

• 2n + 1

• 2n -1