Keystone Algebra: Functions and Data Questions and Answers 1 — Questions and Answers
Question 1: Which of the following sets of ordered pairs represents a function?
- {(-2, 2), (-1, 5), (0, 8), (1, 5), (2, 2)} (Correct answer)
- {(3, 4), (5, -1), (3, 7), (8, 0)}
- {(1, 1), (2, 2), (1, -1), (2, -2)}
- {(-4, 9), (-4, 3), (0, 0), (4, 3)}
Correct answer: {(-2, 2), (-1, 5), (0, 8), (1, 5), (2, 2)}
A function is a relation where each input (x-value) corresponds to exactly one output (y-value). In the correct set, each x-value is unique and paired with only one y-value. The other options show at least one x-value paired with multiple y-values, which violates the definition of a function.
Question 2: A plumber charges a flat fee of $50 for a service call, plus an additional $75 per hour of labor. This relationship is modeled by the function C(h) = 75h + 50, where C(h) is the total cost and h is the number of hours of labor. What do the slope and y-intercept represent in this context?
- The slope is the flat fee, and the y-intercept is the hourly rate.
- The slope is the number of hours, and the y-intercept is the total cost.
- The slope is the hourly rate, and the y-intercept is the flat fee. (Correct answer)
- The slope is the total cost, and the y-intercept is the number of hours.
Correct answer: The slope is the hourly rate, and the y-intercept is the flat fee.
In the linear function C(h) = 75h + 50, which is in the form y = mx + b, the slope 'm' represents the rate of change, and the y-intercept 'b' represents the initial value. Here, the slope is 75, which is the cost per hour ($75 hourly rate). The y-intercept is 50, which is the initial cost before any hours are worked (the $50 flat fee).
Question 3: The table below shows some values for a function, f(x). Which of the following statements best describes this function? | x | f(x) | |---|---| | -2| 4 | | -1| 1 | | 0 | 0 | | 1 | 1 | | 2 | 4 |
- It is a linear function because the x-values increase at a constant rate.
- It is not a function because the f(x) values of 1 and 4 are repeated.
- It is a nonlinear function because the rate of change is not constant. (Correct answer)
- It is a linear function because it passes the vertical line test.
Correct answer: It is a nonlinear function because the rate of change is not constant.
A function is linear if the rate of change (slope) between any two points is constant. From x=-2 to x=-1, the change in f(x) is 1-4 = -3. From x=-1 to x=0, the change is 0-1 = -1. Since the rate of change is not constant, the function is nonlinear. The repetition of f(x) values does not disqualify it from being a function.
Question 4: What is the domain of the function f(x) = √(x - 3)?
- All real numbers
- x ≥ 3 (Correct answer)
- x > 3
- x ≤ 3
Correct answer: x ≥ 3
The domain of a function is the set of all possible input values (x-values) for which the function is defined. The square root function is only defined for non-negative numbers. Therefore, the expression inside the square root, (x - 3), must be greater than or equal to zero. Solving the inequality x - 3 ≥ 0 gives x ≥ 3.
Question 5: A researcher is tracking the population of a certain species of bird. The population started at 80 and has been decreasing by 6 birds each year. Which function, P(t), correctly models the bird population after t years?
- P(t) = 80t - 6
- P(t) = 6t - 80
- P(t) = 80 + 6t
- P(t) = 80 - 6t (Correct answer)
Correct answer: P(t) = 80 - 6t
The initial population is the starting value, which corresponds to the y-intercept in a linear model. The rate of change, or slope, is a decrease of 6 birds per year, so it should be represented as -6. The function should model the initial population of 80 and subtract 6 for each year (t) that passes. Therefore, the correct function is P(t) = 80 - 6t.
Question 6: Which of the following graphs represents y as a function of x?
- A vertical line
- A horizontal line (Correct answer)
- A circle
- A sideways parabola opening to the right
Correct answer: A horizontal line
To determine if a graph represents a function, use the Vertical Line Test. If any vertical line can be drawn that intersects the graph at more than one point, the graph does not represent a function. A horizontal line will pass the vertical line test, as any vertical line will cross it only once. A vertical line, a circle, and a sideways parabola will all fail the vertical line test.
Keystone Algebra: Functions and Data Questions and Answers 1
Which of the following sets of ordered pairs represents a function?