Identifying Functions
Consider the set of ordered pairs:
(2, 5), (3, 7), (2, 8), (5, 10).
Does this relation define a function?
A. Yes
B. No
No. The input value 2 is paired with two different outputs (5 and 8), so it is not a function.
Given the equation y = 3x - 1, is it linear or nonlinear?
A. Linear
B. Nonlinear
Linear; it can be written in slope-intercept form with slope 3 and intercept -1.
For the linear function y = 5x + 10, what is the rate of change and what is the initial value?
A. Rate of change 5; initial value 10
B. Rate of change 5; initial value 0
C. Rate of change 10; initial value 5
D. Rate of change 10; initial value 0
Rate of change 5; initial value 10.
A bike rental costs $10 per hour with no upfront fee.
Let x be the number of hours rented and y be the total cost.
Which equation models this cost?
A. y = 10x
B. y = x + 10
C. y = 10x + 1
D. y = 10 + x
y = 10x
Consider the functions:
A. f(x) = 2x + 1
B. g(x) = x + 3
Which function increases more quickly as x increases?
y = 2x + 1 increases more quickly because its slope 2 is larger than slope 1.
Consider the table:
x: -1, 0, 1
y: 2, 2, 2
Does this relation define a function?
A. Yes
B. No
Yes. Each x-value has exactly one y-value, so it is a function (a constant function).
Consider the table:
x: 0, 2, 4
y: 3, 7, 11
Is this relationship linear or nonlinear?
A. Linear
B. Nonlinear
Linear; the y-values increase by 4 each time x increases by 2, so the rate of change is constant.
A membership program charges a $30 enrollment fee and $8 for each session.
What is the rate of change and what is the initial value?
A. $8 per session; $30 initial
B. $8 per session; $0 initial
C. $30 per session; $8 initial
D. $30 per session; $0 initial
Rate of change $8 per session; initial value $30.
A ride-sharing service charges a $3 booking fee and $0.50 per mile driven. Let x be miles driven and y be total cost.
Which linear equation models this situation?
A. y = 0.50x + 3
B. y = 3x + 0.50
C. y = 0.50x + 0
D. y = 3x + 5
y = 0.50x + 3
Two subscription plans have functions:
A. Plan A: y_1 = 5x + 10
B. Plan B: y_2 = 3x + 20
where x is the number of months and y is total cost.
After how many months will the cost of both plans be equal?
Select one:
A. 4 months
B. 5 months
C. 6 months
D. They will never be equal
After 5 months; solving 5x + 10 = 3x + 20 gives x = 5.
A line crosses the x-axis at x=4 and goes straight up and down.
Does this graph represent a function?
A. Yes
B. No
No; a vertical line has one x-value associated with many y-values, so it's not a function.
Given the table:
x: 0, 1, 2
y: 2, 3, 6
Is this relationship linear or nonlinear?
A. Linear
B. Nonlinear
Nonlinear; the change in y is not constant (1 then 3).
Given two points (1, 4) and (3, 8) on a linear function,
determine the rate of change and the initial value.
A. Slope 2; initial value 2
B. Slope 2; initial value 0
C. Slope 4; initial value 2
D. Slope 4; initial value 0
The slope is 2 and the initial value (y-intercept) is 2.
Vicky's phone is at 100% battery and loses 5% per hour once unplugged. Let x be the number of hours and y be the battery percentage.
Which equation models this situation?
A. y = 100 - 5x
B. y = 5x - 100
C. y = 100x - 5
D. y = 5 - 100x
y = 100 - 5x
Functions f(x) = -3x + 6 and g(x) = 2x - 4.
Which function decreases as x increases?
A. f(x)
B. g(x)
f(x) decreases because its slope is negative (-3); g(x) increases because its slope is positive.
Consider the following equations:
1. y = 4
2. x = -2
3. x^2 + y = 0
Which of these represent y as a function of x? Provide the number(s) for the functions.
Only (1) y = 4 is a function of x. (2) is a vertical line and (3) yields two outputs for some x, so they are not functions.
Consider the function defined by the table below:
x: 1 2 3
y: 4 8 12
Is this relationship linear or nonlinear? If it is linear, write the equation of the line in the form y = mx + b.
Linear. The ratio of change in y (4) to change in x (1) is constant, so the points lie on a straight line. The slope is 4 and the initial value (y-intercept) is 0, so the linear function is y = 4x.
A line passes through the points (0, –2) and (4, 6). Determine its rate of change and initial value.
The slope (rate of change) is 2 and the initial value (y-intercept) is –2.
An online streaming service charges a $20 activation fee and $8 per month for access.
Let x be the number of months and y be the total cost.
Write a linear equation modeling this situation, and state the rate of change and initial value.
The equation is y = 8x + 20. The rate of change is $8 per month, and the initial value (activation fee) is $20.
Function f is defined by the points (0, 2), (1, 4), (2, 6).
Function g is defined by g(x) = 3x + 1.
Determine the rate of change and initial value for each function.
Which function increases faster and which has the larger starting value?
For f, the rate of change is 2 and the initial value is 2. For g(x) = 3x + 1, the rate of change is 3 and the initial value is 1. Therefore g increases faster, but f starts with a higher initial value.
Consider the following relations:
A. x^2 + y^2 = 16
B. y = 3x^2 + 2
C. y^2 = 4x
Which of these represent y as a function of x? Provide the letter(s) for the functions.
Only relation B is a function of x. Relations A and C fail because for some x-values there are two corresponding y-values. In y = 3x^2 + 2, each input x produces exactly one output y.
Kara records the distance she has cycled at different times:
- After 1 hour she has traveled 15 miles,
- after 2 hours 31 miles,
- and after 3 hours 48 miles.
Is this relationship linear or nonlinear?
It is nonlinear. The increments in distance over each hour (from 15 to 31 and from 31 to 48) are 16 and 17 miles, respectively, so the rate of change is not constant. A linear relationship would have the same increase per hour.
During an experiment, the temperature of a liquid was recorded:
10 minutes after boiling, the temperature was 90°C,
and 20 minutes after boiling, it was 70°C.
Assuming the temperature decreases linearly over time, determine the rate of change and the initial temperature at time 0,
and write an equation modeling the temperature y as a function of time x.
The slope (rate of change) is (70 - 90) °C divided by (20 - 10) minutes = -2 °C per minute. Using 90 = -2(10) + b gives b = 110, so the initial temperature is 110 °C. The equation is y = -2x + 110.
A water tank contains 100 gallons of water.
After 12 hours of draining at a constant rate, it holds 40 gallons.
Let x be the number of hours since draining began and y be the gallons of water left.
Determine the rate of change and the initial value, and write a linear equation modeling this situation.
The tank loses 60 gallons over 12 hours, so the rate of change is -5 gallons per hour. The initial value at x = 0 is 100 gallons. The linear model is y = -5x + 100.
Function f is represented by the table:
x: 0, 2, 4
f(x): 10, 7, 4
Function g is defined by g(x) = -2x + 12.
Determine the rate of change and initial value for each function.
Which function decreases faster, and which has the higher initial value?
From the table, f drops from 10 to 7 to 4 as x increases by 2, so its slope is (4–10)/(4–0) = -6/4 = -1.5 and the initial value is 10 at x = 0. For g(x) = -2x + 12, the slope is -2 and the initial value is 12 at x = 0. Since -2 < -1.5, g decreases faster (its rate of change is more negative), and g also starts higher because 12 > 10.